UC-NRLF 


SB    320    EDO 


IN   MEMORIAM 
FLOR1AN  CAJORI 


ORIGINAL  EXERCISES 


IN 


PLANE  AND  SOLID  GEOMETRY 


BY 


LEVI   L.   CONANT,  Ph.D. 

PROFESSOR   OF  MATHEMATICS   IN  THE  WORCESTER 
POLYTECHNIC   INSTITUTE 


>>Kc 


NEW  YORK  •:•  CINCINNATI  •:•  CHICAGO 

AMERICAN    BOOK    COMPANY 


(263 


COPYRIGHT,    1905,   BY 

LEVI  L.  CONANT. 
Entered  at  Stationers'  Hall,  London. 

conant'b  ex.  geom. 
W.  P.    1 

cajori 


PREFACE 

During  the  last  few  years  the  custom  has  become  quite 
general  in  high  schools  and  academies  of  giving  a  general 
review  of  preparatory  mathematics  during  the  last  year  of 
the  course ;  and  of  that  review  a  portion  now  universally 
recognized  as  holding  a  place  of  great  importance  is  the 
original  work  designed  to  round  out  a  student's  course  in 
plane  and  solid  geometry.  This  collection  of  theorems, 
constructions,  and  numerical  problems  is  designed  to  sup- 
ply the  material  for  this  original  work. 

The  exercises  which  make  up  the  book  are  arranged 
somewhat  promiscuously,  the  design  being  that  the  stu- 
dent shall  not  be  restricted  in  his  method  of  proof  ;  just 
as  he  is  not  restricted  in  his  method  of  proof  for  any 
original  exercise  he  may  be  asked  to  solve  when  he 
attempts  to  pass  his  entrance  examination  to  college. 

In  the  preparation  of  these  exercises  the  author  has 
sought : — 

1.  To  obtain  variety,  combined  with  proper  gradation 
from  easy  to  difficult  problems. 

2.  To  generalize  whenever  it  was  possible.  In  other 
words,  to  gather  up  in  one  problem  as  many  kindred 
problems  as  seemed  judicious. 

918254 


4  PREFACE 

3.  To  interest  the  student  in  the  history  of  geometry. 

4.  To  teach  from  time  to  time  new  principles. 

The  author  has  solved  all  the  problems,  to  test  their 
fitness  for  class  work. 

For  assistance  in  the  preparation  of  the  work  the  author 
desires  to  acknowledge  his  indebtedness  to  Mr.  J.  W. 
Thomas,  formerly  instructor  in  mathematics  in  the  Wor- 
cester Polytechnic  Institute. 

LEVI  L.   CONANT. 


CONTENTS 


PAGE 


I.  Triangles  and  Quadrilaterals 7 

II.  The  Circle 21 

III.  Constructions  .  47 

IV.  Similar  Figures •        •        .68 

V.  Areas •        •        •        .84 

VI.  Miscellaneous  Theorems  and  Problems  ...      96 

VII.  Theorems    and    Problems    relating    to    Common 

Geometrical  Solids 114 


ORIGINAL  EXERCISES  IN  PLANE  AND 
SOLID    GEOMETRY 

I.    TRIANGLES   AND   QUADRILATERALS 

1.  The  medians  drawn  from  the  equal  angles  of  an 
isosceles  triangle  are  equal. 

Is  this  true  of  any  other  lines  except  the  medians  ?  Are 
the  medians  ever  equal  except  in  the  case  of  the  isosceles 
triangle  ? 

2.  Any  altitude  of  a  triangle  is  less  than  half  the  sum 
of  the  adjacent  sides ;  and  the  sum  of  the  altitudes  is  less 
than  the  perimeter. 

Is  this  true  also  of  the  medians  ?  of  the  angle  bisectors  ? 

3.  Any  line  drawn  through  the  intersection  of  the  diago- 
nals of  a  parallelogram,  and  terminated  by  the  perimeter, 
bisects  the  parallelogram  and  also  its  perimeter,  and  is 
itself  bisected  by  the  intersection  of  the  diagonals. 

Note.  The  intersection  of  the  diagonals  of  a  parallelogram  is 
called  its  center.  Give  a  general  definition  of  the  term  center  of  a 
figure. 

4.  A  line  drawn  between  two  sides  of  a  triangle,  parallel 
to  the  base  of  the  triangle  and  equal  to  half  the  base,  bisects 
each  of  the  other  sides. 

What  other  lines  does  this  line  bisect  ? 

5.  The  medians  of  a  triangle  meet  in  a  point ;  and  this 
point  divides  each  median  into  two  parts  such  that  the 
greater  is  twice  the  less. 

7 


8  TRIANGLES  AND  QUADRILATERALS 

Note.  The  intersection  of  the  medians  of  a  triangle  is  called  the 
centroid  of  the  triangle.  It  is,  also,  sometimes  called  the  center  of 
gravity  of  the  triangle. 

6.  The  diagonals  of  an  isosceles  trapezoid  are  equal. 

7.  State  and  prove  the  converse. 

If  a  theorem  is  true,  is  its  converse  necessarily  true? 
Give  an  example  of  this. 

8.  If  any  number  of  points,  n,  are  so  situated  that  not 
more  than  two  of  them  lie  on  the  same  straight  line,  the 
number  of  lines  that  can  be  drawn  connecting  them,  pair 

by  pair,  is  |(w-l). 

9.  The  maximum  number  of  points  of  intersection  of 

n  straight  lines  is  -(n—Y). 

A 

A  comparison  of  the  last  two  theorems  will  show  that 
either  may  be  transformed  into  the  other  by  a  slight  change 
in  the  wording.  The  lines  of  the  former  are  replaced  by 
the  points  of  the  latter,  and  the  points  of  the  former  by 
the  lines  of  the  latter.  This  is  an  illustration  of  what  is, 
in  mathematics,  known  as  the  law  of  duality,  a  principle 
of  the  greatest  importance  in  certain  branches  of  that 
science. 

10.  If  either  leg  of  an  isosceles  triangle  be  produced 
from  the  vertex  by  its  own  length,  and  the  extremity 
joined  to  the  extremity  of  the  base,  the  joining  line  is 
perpendicular  to  the  base. 

How  much  larger  is  the  right  triangle  than  the  isosceles 
triangle  ? 

11.  The  square  described  on  the  diagonal  of  a  given 
square  is  twice  that  square. 


TRIANGLES  AND   QUADRILATERALS  9 

12.  Divide  a  straight  line  so  that  the  square  on  one  of 
its  segments  shall  be  double  the  square  on  the  other. 

13.  The  bisector  of  one  of  the  base  angles  of  an  isosceles 
triangle  makes  with  the  opposite  leg  an  angle  of  61°  40'. 
Find  all  the  angles  of  the  triangle. 

14.  If  through  any  point  in  the  base  of  an  isosceles 
triangle  lines  are  drawn  parallel  to  the  legs,  a  parallelo- 
gram is  formed  whose  perimeter  is  equal  to  the  sum  of 
the  legs  of  the  triangle. 

15.  The  lines  drawn  from  either  pair  of  opposite  ver- 
tices of  a  parallelogram  to  the  middle  points  of  the  parallel 
sides  lying  opposite  the  vertices  from  which  they  are 
drawn,  trisect  the  diagonal  connecting  the  other  pair  of 
vertices. 

16.  Two  straight  lines  from  two  vertices  of  a  triangle 
to  the  opposite  sides  respectively,  cannot  bisect  each  other, 
unless —  . 

Let  the  student  complete  and  demonstrate  the  above. 

17.  What  is  the  area  of  an  equilateral  triangle  whose 
side  is  a  ? 

18.  What  is  the  area  of  an  equilateral  triangle  whose 
altitude  is  h  ? 

19.  Prove  algebraically  that  (i)  if  m  be  an  even  number, 
then  m,  \  m2  —  1,  \m2  +  1  are  numerically  the  sides  of  a 
right  triangle ;  (ii)  if  n  be  an  odd  number,  then  n,  \(n2  —  1), 
i(w2-|-l)  fulfill  the  same  condition. 

The  first  part  of  the  above  theorem  is  due  to  the  Greek 
philosopher  Plato,  and  the  second  to  Pythagoras. 

20.  The  angle  formed  by  the  bisectors  of  two  exterior 
angles  of  a  triangle  is  equal  to  a  right  angle  minus  half 
the  third  angle  of  the  triangle. 


10  TRIANGLES   AND  QUADRILATERALS 

21.  How  many  hexagons  may  be  made  to  touch  at  one 
point  ?  How  many  regular  hexagons  ?  How  many  equal 
regular  hexagons  ?     State  the  reason  in  each  case. 

22.  The  perpendicular  bisector  of  the  base  of  an 
isosceles  triangle  passes  through  the  vertex,  and  bisects 
the  vertical  angle. 

Which  is  the  easier  proof  for  this  proposition,  the 
direct  or  the  indirect  ?     What  is  an  indirect  proof  ? 

23.  The  bisector  of  any  angle  of  a  triangle  lies  between 
the  altitude  and  the  median  drawn  from  that  vertex. 

What  is,  then,  always  true  of  the  relative  magnitudes 
of  these  three  lines? 

24.  If  the  hypotenuse  of  a  right  triangle  be  twice  the 
shorter  leg,  one  of  the  acute  angles  is  twice  the  other. 

25.  If  through  the  vertex  of  an  angle  of  a  parallelogram 
any  straight  line  be  drawn,  the  perpendicular  to  it  from 
the  vertex  of  the  opposite  angle  is  equal  to  the  sum  or 
difference  of  the  perpendiculars  to  it  from  the  vertices 
of  the  other  two  angles,  according  as  the  straight  line  lies 
without  the  parallelogram  or  intersects  it. 

26.  A  line  connecting  the  middle  points  of  any  two 
sides  of  a  triangle  is  parallel  to  the  third  side  and  equal 
to  half  that  side. 

27.  If  the  diagonals  of  a  parallelogram  are  equal,  the 
parallelogram  is  a  rectangle. 

Is  a  square  a  rectangle  ?  Is  a  rhombus  a  parallelogram  ? 
Is  a  square  a  rhombus  ? 

28.  If  ABO  be  any  triangle,  and  AD  the  bisector  of  the 
exterior  angle  between  BA  and  CA  produced,  meeting  CB 
produced  in  D,  then  AB  •  AO  =  DB  •  DO  -  AD*. 


TRIANGLES   AND   QUADRILATERALS  11 

Can  this  line  ever  be  equal  to  the  median  of  the 
trapezoid  ? 

29.  The  angle  formed  by  the  bisector  of  one  of  the 
base  angles  of  a  triangle  and  the  bisector  of  the  exterior 
angle  at  the  other  extremity  of  the  base  is  equal  to  half 
the  vertical  angle  of  the  triangle. 

30.  The  angle  formed  by  the  altitude  and  the  angle 
bisector  drawn  from  the  same  vertex  of  the  triangle  is 
equal  to  half  the  difference  of  the  two  remaining  angles 
of  the  triangle. 

Is  this  theorem  true  when  the  altitude  falls  outside  the 
triangle  ? 

31.  If  from  the  right  angle  of  a  right  triangle  the 
altitude,  the  median,  and  the  angle  bisector  be  drawn, 
the  bisector  is  also  the  bisector  of  the  angle  formed  by 
the  other  two. 

What  is  the  value  of  this  angle  in  terms  of  the  acute 
angles  of  the  original  triangle  ? 

32.  The  greatest  of  the  angle  bisectors  of  a  triangle 
falls  on  the  least  side. 

33.  What  is  the  side  of  a  regular  hexagon  whose  area 
igji? 

34.  The  bisector  of  either  of  the  base  angles  of  an 
isosceles  triangle  whose  vertical  angle  is  36°  divides  the 
triangle  into  two  isosceles  triangles. 

35.  The  sum  of  the  perpendiculars  drawn  from  any 
point  within  a  regular  polygon  of  n  sides  to  each  of  these 
sides  is  constant,  and  is  equal  to  n  times  the  apothem. 

36.  Into  how  many  equal  parts  can  a  rectangular  strip 
of  paper  be  divided  by  one  cut  of  a  knife  or  a  pair  of 
scissors  ? 


12  TRIANGLES  AND  QUADRILATERALS 

37.  The  perpendiculars  from  two  vertices  of  a  triangle 
to  the  opposite  sides  divide  each  other  into  segments 
which  are  reciprocally  proportional. 

Is  there  any  modification  of  this  theorem  in  the  case  of 
the  isosceles  or  the  equilateral  triangle  ? 

38.  Construct  a  square  whose  area  shall  be  three  times 
the  area  of  a  given  square. 

39.  The  bisectors  of  the  angles  of  a  quadrilateral  form 
another  quadrilateral  whose  opposite  angles  are  supple- 
mentary.    Consider  each  of  the  six  kinds  of  quadrilaterals. 

40.  ABC  is  an  equilateral  triangle  in  which  BB  is  per- 
pendicular to  AC,  and  BE  is  perpendicular  to  BC ;  then 
is  EC  equal  to  one  third  BE. 

41.  The  lines  joining  the  middle  points  of  the  sides  of 
any  quadrilateral,  taken  in  order,  form  a  parallelogram 
whose  perimeter  is  equal  to  the  sum  of  the  diagonals  of 
the  quadrilateral.     Examine  each  kind  of  quadrilateral. 

42.  Three  angles  of  a  quadrilateral  are  80°,  85°,  110° ; 
construct  a  quadrilateral,  none  of  whose  angles  shall  equal 
any  angle  of  this  quadrilateral. 

43.  If  a  line  be  drawn  from  the  angle  at  one  vertex  of 
a  triangle  perpendicular  to  the  bisector  of  another  angle, 
and  through  their  intersection  a  line  be  drawn  parallel  to 
the  side  opposite  the  first  angle,  the  line  last  drawn  will 
bisect  each  of  the  other  two  sides. 

Hint.  Continue  the  perpendicular  till  it  intersects  the  opposite 
side  or  the  opposite  side  produced. 

44.  The  bisector  of  the  angle  C  of  the  triangle  ABC 
meets  AB  in  D,  and  BE  is  drawn  parallel  to  AC  meeting 
BC  in  E  and  the  bisector  of  the  exterior  angle  at  Cm  F; 
prove  BE=EF. 


TRIANGLES  AND  QUADRILATERALS  13 

45.  If  two  of  the  medians  of  a  triangle  be  produced 
through  the  respective  sides  to  which  they  are  drawn, 
each  by  its  own  length,  the  line  joining  their  external 
extremities  will  pass  through  one  of  the  vertices  of  the 
triangle. 

46.  From  the  vertices  of  a  parallelogram  ABCB  perpen- 
diculars are  drawn,  meeting  the  diagonals  AC  and  BD  in 
U,  F,  G-,  Hi  respectively ;  then  is  EFGLH  a  parallelogram 
similar  to  ABCB.     Define  similar  parallelograms. 

47.  If  lines  be  drawn  from  the  vertices  of  a  square, 
bisecting  the  opposite  sides  in  order,  a  second  square 
will  be  formed  whose  area  is  \  the  area  of  the  first 
square. 

Can  a  square  be  formed  in  a  similar  manner  within  any 
other  kind  of  parallelogram  ? 

48.  How  many  braces  does  it  take  to  stiffen  a  three- 
sided  plane  frame?  A  four-sided  frame?  A  five-sided 
frame  ? 

49.  If  a  pavement  is  to  be  laid  with  blocks  whose  upper 
faces  are  equal,  regular  polygons,  these  faces  must  be 
triangles,  squares,  or  hexagons. 

50.  If  the  pavement  is  to  be  laid  with  blocks  of  two 
different  kinds  at  each  angular  point,  the  upper  faces 
being  regular  polygons  with  sides  of  the  same  length,  it 
can  be  laid  only  with :  — 

(i)    Triangles  and  squares. 

(ii)    Triangles  and  hexagons, 
(iii)    Triangles  and  dodecagons, 
(iv)    Squares  and  octagons. 

51.  If  the  pavement  is  to  be  laid  with  blocks  of  three 
different   kinds   at   each  angular  point,  the  upper  faces 


14  TRIANGLES  AND  QUADRILATERALS 

being  regular  polygons  with  sides  of  the  same  length,  it 
can  be  laid  only  with  :  — 

(i)    Triangles,  squares,  and  hexagons, 
(ii)    Squares,  hexagons,  and  dodecagons. 

52.  The  sum  of  the  perpendiculars  let  fall  from  any 
point  in  the  base  of  an  isosceles  triangle  upon  the  legs  is 
constant. 

What  if  the  perpendiculars  are  let  fall  from  a  point  in 
the  base  produced  ?  How  may  this  theorem  be  extended 
in  the  case  of  equilateral  triangles  ? 

53.  The  diagonals  of  a  trapezoid  divide  each  other  into 
segments  that  are  reciprocally  proportional. 

54.  The  cross  section  of  a  bee's  cell  is  a  regular  hexagon. 
Could  any  other  form  be  used  to  so  good  advantage  ? 
Why? 

The  problem  of  the  bee  is  to  store  away  a  maximum 
amount  of  honey  while  using  a  minimum  amount  of  wax. 
The  above  question  must,  then,  be  studied  with  reference 
to  both  these  conditions. 

55.  The  sum  of  the  diagonals  of  a  quadrilateral  is  less 
than  its  perimeter,  but  greater  than  half  its  perimeter. 

56.  Is  there  a  point  within  a  triangle  such  that  the 
area  of  the  figure  will  be  bisected  by  every  line  drawn 
through  it  ?  By  any  one  line  ?  By  more  than  one 
line? 

57.  The  lines  which  join  the  middle  points  of  the  sides 
of  a  triangle  form  a  triangle  whose  area  is  one  fourth  the 
area  of  the  original  triangle. 

58.  The  area  of  a  triangle  having  an  angle  of  30°  is 
one  fourth  the  product  of  the  sides  including  that  angle. 


TRIANGLES  AND  QUADRILATERALS  15 

59.  The  altitude  dropped  from  the  vertex  upon  the 
base  of  any  isosceles  triangle  whose  vertical  angle  is  120° 
is  greater  than  half  of  either  leg. 

60.  A  line  joining  the  middle  points  of  the  parallel 
sides  of  a  trapezoid  passes  through  the  intersection  of 
the  diagonals,  and  also  through  the  intersection  of  the 
non-parallel  sides  produced. 

61.  The  sum  of  the  perpendiculars  dropped  from  any 
point  within  an  equilateral  triangle  upon  the  three  sides 
is  constant.     To  what  is  it  equal  ? 

What  if  the  point  be  without  the  triangle? 

62.  A  line  joining  the  middle  points  of  the  non-parallel 
sides  of  a  trapezoid  bisects  each  of  the  diagonals. 

Does  it  bisect  anything  else  ? 

63.  Three  given  straight  lines  meet  at  a  point;  draw 
another  straight  line  so  that  the  two  portions  of  it  inters 
cepted  between  the  given  lines  may  be  equal  to  one 
another. 

How  can  a  second  solution  to  this  problem  be  obtained  ? 

64.  A  perpendicular  dropped  from  either  extremity  of 
the  base  of  an  isosceles  triangle  upon  the  opposite  leg 
forms  with  the  base  an  angle  equal  to  half  the  vertical 
angle  of  the  triangle. 

65.  Each  of  the  lines  joining  the  middle  points  of  the 
opposite  sides  of  a  quadrilateral  bisects  the  line  joining 
the  middle  points  of  the  diagonals  of  the  quadrilateral. 

Hint.  The  line  joining  the  opposite  sides  and  the  line  joining  the 
middle  points  of  the  diagonals  are  the  diagonals  of  a  parallelogram. 

66.  If  one  angle  of  a  triangle  be  double  or  triple 
another,  the  triangle  can  be  divided  into  two  isosceles 
triangles. 


16  TRIANGLES  AND  QUADRILATERALS 

67.  Lines  drawn  through  the  vertices  of  a  triangle, 
parallel  respectively  to  the  opposite  sides,  enclose  a  tri- 
angle four  times  as  large  as  the  first  triangle  ;  and  the 
sides  of  the  second  triangle  are  bisected  by  the  vertices 
of  the  first. 

68.  In  the  triangle  ABC,  a  =  25,c  =  16,  and  the  median 
on  b  is  12  ;  find  b. 

69.  If  the  square  corner  of  a  sheet  of  paper  be  folded 
any  number  of  times  so  that  the  consecutive  lines  of 
folding  are  all  parallel,  and  equidistant  from  each  other, 
the  consecutive  areas  thus  obtained  will  be  to  each  other 
as  1  :  3  :  5  :  7,  etc. 

70.  The  altitudes  of  a  triangle  are  to  each  other  in  the 
inverse  ratio  of  the  sides  on  which  they  fall. 

If  two  sides  of  a  triangle  are  6  in.  and  8  in.  respectively, 
how  much  can  be  determined  about  the  altitudes  upon 
them? 

71.  The  sum  of  the  four  lines  drawn  to  the  vertices  of 
any  quadrilateral  from  any  point  except  the  intersection 
of  the  diagonals  is  greater  than  the  sum  of  the  diagonals. 

72.  The  line  joining  the  middle  points  of  the  diagonals 
of  a  trapezoid  is  equal  to  half  the  difference  of  the  parallel 
sides. 

73.  If  the  line  joining  the  middle  points  of  the  diagonals 
of  a  trapezoid  be  extended  to  meet  the  non-parallel  sides, 
the  sum  of  the  two  remaining  segments  of  this  line  is 
equal  to  the  shorter  of  the  two  parallel  sides  of  the 
trapezoid. 

74.  If  any  point  within  a  triangle  be  joined  to  the 
three  vertices,  the  sum  of  the  joining  lines  is  less  than 
the  perimeter  but  greater  than  half  the  perimeter. 


TRIANGLES  AND  QUADRILATERALS  17 

Is  this  theorem  still  true  if  the  point  be  taken  on  the 
perimeter  of  the  triangle? 

75.  If  from  the  acute  angle  A  of  a  right  triangle  ABO, 
having  its  right  angle  at  0,  a  line  be  drawn  cutting  BO  in 
2),  will  AB2  +  CD2  =  AD2  +  OB2  ? 

76.  What  is  the  area  of  an  equilateral  triangle  whose 
center  of  gravity  is  6  in.  from  the  vertex  ? 

77.  The  bisector  of  any  angle  of  a  triangle  and  the 
bisectors  of  the  exterior  angles  at  the  other  two  vertices 
are  concurrent. 

78.  The  altitude  of  a  trapezoid  is  A,  and  its  parallel 
sides  are  a  and  b,  with  a<b;  find  the  area  of  the  triangle 
formed  by  a  and  the  non-parallel  sides  produced. 

What  is  the  area  if  h  =  4  in.,  a  =  6  in.,  b  =  8  in.  ? 

79.  If  ABO  be  a  reentrant  angle  of  a  concave  quadri- 
lateral ABCB,  prove  that  the  angle  ABO,  exterior  to  the 
figure,  is  equal  to  the  sum  of  the  interior  angles  A,  B,  O. 

80.  If  any  point  E  be  connected  with  the  vertices  of  a 
rectangle  ABCB,  then  is  AE2  +  OE2  =  BE2  +  BE2. 

81.  The  diagonals  of  a  trapezoid  are  10  in.  and  15  in.  re- 
spectively, and  cut  each  other  so  that  the  segments  of  the  lat- 
ter are  6  in.  and  9  in. ;  what  are  the  segments  of  the  former? 

82.  On  a  piece  of  paper  of  given  size  two  lines  are 
drawn  whose  intersection  lies  off  the  paper ;  show  how  to 
draw  a  line  through  any  given  point  on  the  paper  so  that 
it  would,  if  produced,  pass  through  the  intersection  of  the 
first  two  lines. 

83.  ABOB  is  any  parallelogram,  and  E  is  any  point  in 
the  diagonal  A  0  produced ;  prove  that  the  triangles  EBO 
and  EBO  are  equal  in  area. 

conant's  ex.  geom.  — 2 


18  TRIANGLES  AND  QUADRILATERALS 

84.  Bisect  a  parallelogram  by  a  line  drawn  through  a 
given  point,  either  within  or  without  the  parallelogram. 

85.  What  is  the  area  of  a  trapezoid  whose  parallel  sides 
are  25  and  35,  and   whose  non-parallel  sides  are  5  and 

V65? 

86.  What  part  of  a  triangle  is  between  the  base  and  a 
line  parallel  to  the  base,  passing  through  the  center  of 
gravity  of  the  triangle  ? 

87.  The  three  lines  drawn  through  the  intersection  of 
the  bisectors  of  the  angles  of  an  equilateral  triangle, 
parallel  to  the   sides   of   the   triangle,  trisect   the   sides. 

88.  Two  equal  triangles  stand  with  their  bases  on  the 
same  straight  line,  but  with  their  vertices  on  opposite 
sides  of  this  line ;  prove  that  the  line  joining  the  vertices 
of  the  two  triangles  is  bisected  by  the  line  on  which  the 
bases  stand. 

89.  If  two  triangles  have  two  sides  of  the  one  equal 
respectively  to  two  sides  of  the  other,  and  the  included 
angles  supplementary,  the  triangles  are  equal  in  area. 

90.  If  equilateral  triangles  are  described  outwardly  on 
the  three  sides  of  any  triangle,  the  lines  joining  the 
outer  vertices  of  the  equilateral  triangles  to  the  opposite 
vertices  respectively  of  the  given  triangle  cut  each  other 
at  angles  of  120°. 

Are  these  lines  equal  ?     Are  they  concurrent  ? 

91.  ABOD  is  a  parallelogram,  and  E  is  any  point  with- 
out it ;  prove  that  the  sum  or  the  difference  of  the  triangles 
EAB,  ECD  is  equal  in  area  to  half  the  parallelogram. 

When  is  it  the  sum  and  when  the  difference  ? 
Hint.    Draw  through  E  a  line' parallel  to  AB  and  CD. 


TRIANGLES  AND  QUADRILATERALS  19 

92.  A  triangle  is  equal  in  area  to  the  sum  or  the  differ- 
ence of  two  triangles  on  the  same  base  or  on  equal  bases 
if  its  altitude  is  equal  to  the  sum  or  the  difference  of  the 
altitudes  of  the  two  triangles. 

93.  A  parallelogram  is  equal  in  area  to  the  sum  or  the 
difference  of  two  parallelograms  of  the  same  or  of  equal 
altitudes  if  its  base  is  equal  to  the  sum  or  to  the  difference 
of  the  bases  of  the  two  parallelograms. 

94.  The  perpendiculars  from  the  vertices  of  an  acute- 
angled  triangle  to  the  opposite  sides  are  the  bisectors  of 
the  angles  of  the  triangle  formed  by  joining  the  feet  of 
these  perpendiculars. 

What  modification  does  this  theorem  undergo  in  the  case 
of  the  obtuse-angled  triangle  ? 

95.  A  triangle  having  two  sides  equal  respectively  to 
the  diagonals  of  a  quadrilateral,  and  the  included  angle 
equal  to  either  angle  between  these  diagonals,  has  the  same 
area  as  the  quadrilateral. 

Hint.  From  two  opposite  vertices  of  the  quadrilateral  drop 
perpendiculars  to  the  diagonal  connecting  the  other  two  vertices, 
and  find  the  area  of  each  of  the  four  triangles  in  the  quadrilateral 
in  terms  of  these  perpendiculars  and  segments  of  the  diagonals.  The 
sum  of  these  will  equal  the  area  of  the  given  triangle. 

96.  ABCD  is  a  parallelogram,  and  E  is  any  point  with- 
out the  angle  BAD  and  its  opposite  vertical  angle ;  then 
is  the  triangle  EAC  equal  in  area  to  the  sum  of  the 
triangles  EAT)  and  EAB.     (See  No.  92.) 

97.  In  the  preceding  theorem,  if  the  point  E  be  within 
the  angle  BAD  or  its  opposite  vertical  angle,  the  triangle 
EAQ  is  equal  in  area  to  the  difference  between  the  tri- 
angles EAD  and  EAB.     (No.  92.) 


20  TRIANGLES  AND  QUADRILATERALS 

Hint.  Extend  AE;  through  C  draw  a  line  parallel  to  AE.  The 
altitude  of  A  EC  can  then  be  proved  equal  to  the  difference  of  the 
altitudes  of  the  other  two  triangles. 

98.  ABOD  is  a  parallelogram,  and  through  E,  any  point 
within  it,  lines  are  drawn  parallel  to  the  sides  of  the  par- 
allelogram; then  is  the  difference  of  the  parallelograms 
BE,  DE  equal  in  area  to  twice  the  triangle  A  CE.  (See 
No.  97.) 

99.  ABO  is  a  triangle,  and  D  is  any  point  in  AB;  draw 
through  D  a  line  DE  to  meet  BO  produced  in  E,  so  that 
the  triangles  DBE  and  ABO  shall  be  equal  in  area. 

100.  On  the  sides  of  the  right  triangle  ABO,  right 
angled  at  C,  construct  the  squares  A  ODE,  BOFK,  ABLE, 
outwardly  from  the  triangle.  Draw  ON  parallel  to  BL. 
In  the  figure  thus  formed,  prove:  — 

(i)  AO2  +  OL2  =  OB*  +  OH2. 
(ii)  EH2=B02  +  ±A02. 
(iii)  LK2  +  EH2  =5  AB2. 
(iv)  E,  0,  if  are  collinear. 
(v)  AK  is  perpendicular  to  OL. 


II.     THE   CIRCLE 

101.  If  the  inscribed  and  circumscribed  circles  of  a 
triangle  are  concentric,  the  triangle  is  equilateral. 

What  modification  of  this  theorem  must  be  made  for  the 
isosceles  triangle  ? 

102.  Three  consecutive  sides  of  an  inscribed  quadri- 
lateral subtend  arcs  of  75°,  92°,  104:°,  respectively;  find 
each  angle  of  the  quadrilateral,  and  the  angles  between 
the  diagonals. 

103.  A  triangle  is  formed  by  joining  the  points  of 
contact  of  a  circumscribed  triangle;  prove  that  any  angle 
of  the  inscribed  triangle  is  equal  to  a  right  angle  minus 
half  the  opposite  angle  of  the  circumscribed  triangle. 

104.  What  is  the  locus  (i)  of  the  middle  points  of  a 
system  of  parallel  chords  in  a  circle?  (ii)  Of  the  middle 
points  of  a  system  of  equal  chords  ? 

105.  The  perpendiculars  erected  at  the  middle  points  of 
the  sides  of  an  inscribed  quadrilateral  are  concurrent. 

What  if  it  be  a  pentagon  ?     A  hexagon  ? 

106.  The  line  joining  the  middle  points  of  the  non- 
parallel  sides  of  a  circumscribed  trapezoid  is  equal  to  one 
fourth  the  perimeter. 

What,  then,  is  its  value  in  terms  of  the  non-parallel 
sides  ? 

107.  If  two  of  the  adjacent  sides  of  an  inscribed  quad- 
rilateral subtend  arcs  of  70°  and  110°,  respectively,  and 

21 


22  THE  CIRCLE 

one  of  the  angles  formed  by  the  diagonals  is  95°,  find  each 
of  the  angles  of  the  quadrilateral. 

108.  The  line  joining  the  middle  points  of  the  arcs 
subtended  by  the  sides  of  an  inscribed  angle  A  inter- 
sects the  sides  of  the  angle  in  B,  (7,  respectively;  prove 
AB  =  AC. 

109.  The  sum  of  the  alternate  angles  of  an  inscribed 
octagon  is  equal  to  six  right  angles. 

What  is  the  generalization  of  the  preceding  theorem  for 
any  inscribed  polygon  having  an  even  number  of  sides? 

110.  What  is  the  ratio  between  the  sides  of  an  inscribed 
and  of  a  circumscribed  equilateral  triangle  ? 

Is  the  ratio  the  same  in  the  case  of  the  inscribed  and 
circumscribed  squares  ? 

ill.  The  angle  formed  by  two  tangents  is  equal  to 
twice  the  angle  between  the  chord  of  contact  and  the 
radius  to  the  point  of  contact. 

112.  The  radius  of  a  circle  is  6  in. ;  what  is  the  locus 
of  a  point  4  in.  from  the  circle  ?  6  in.  from  the  circle  ?  8  in. 
from  the  circle  ? 

113.  Find  a  point  equidistant  from  three  intersecting 
lines. 

How  many  solutions  are  there  ? 

114.  If  any  point  on  a  circle  be  joined  to  the  vertices  of 
an  inscribed  equilateral  triangle,  the  greatest  of  the  join- 
ing lines  is  equal  to  the  sum  of  the  other  two. 

Hint.  From  the  point  on  the  circle,  lay  off  on  the  longest  joining 
line  a  segment  equal  to  one  of  the  other  distances,  and  join  the  ex- 
tremity of  the  segment  to  the  extremity  of  that  joining  line  to  which 
it  is  equal. 


THE  CIRCLE  23 

115.  The  radius  of  a  circle  is  4  in. ;  find  the  altitudes 
of  the  inscribed  and  of  the  circumscribed  equilateral  tri- 
angle. 

116.  Two  circles  are  tangent  externally  at  A,  and  a 
common  tangent  touches  the  two  circles  in  B,  C,  respec- 
tively;  prove  that  the  angle  BAG  is  a  right  angle. 

117.  In  the  above  problem,  the  line  of  centers  of  the 
two  given  circles  is  tangent  to  a  third  circle,  whose 
diameter  is  BO. 

118.  One  chord  of  a  circle  is  twice  as  long  as  another ; 
what  is  the  ratio  of  their  distances  from  the  center  of  the 
circle  ? 

119.  If  two  circles  are  tangent  internally,  they  cannot 
have  the  same  center. 

Find  the  locus  of  the  center  of  a  circle  which  has  a 
given  radius  r,  and  which  also  :  — 

120.  Passes  through  a  given  point. 

121.  Touches  a  given  circle,  internally  or  externally. 

122.  Cuts  a  given  line  so  that  the  chord  intercepted  has 
a  given  length. 

123.  Cuts  a  given  circle  in  a  diameter. 

124.  Cuts  a  given  circle  orthogonally. 

125.  Through  a  given  point  within  a  circle  draw  a 
chord  which  shall  be  bisected  by  that  point. 

126.  Two  circles  whose  centers  are  A,  B,  are  tangent, 
either  internally  or  externally,  and  through  their  point  of 
contact  a  line  is  drawn  cutting  the  circles  in  (7,  D  ;  prove 
that  (i)  the  radii  A  (7,  BD,  are  parallel ;  (ii)  the  tan- 
gents drawn  at  (7,  2),  are  parallel. 


24  THE  CIRCLE 

127.  All  circles  which  pass  through  a  fixed  point,  and 
have  their  centers  on  a  given  line,  pass  through  a  second 
fixed  point. 

128.  If  two  equal  chords  of  a  circle  intersect,  the  seg- 
ments of  the  one  are  equal  respectively  to  those  of  the 
other.     This  is  a  special  case  of  what  general  theorem  ? 

129.  If  two  chords  of  a  circle,  drawn  from  the  same 
point  in  the  circumference,  make  equal  angles  with  the 
tangent  at  that  point,  the  chords  are  equal. 

130.  If  two  circles  are  tangent  externally  at  J.,  the  com- 
mon tangent  at  A  bisects  each  of  the  other  common  tangents. 

131.  The  angle  formed  by  the  lines  joining  the  opposite 
points  of  contact  of  the  circumscribed  quadrilateral  ABCD 
is  either  equal  or  supplementary  to  %(A  +  (7).  When  is 
it  equal  and  when  supplementary  ? 

132.  The  sum  of  two  opposite  sides  of  a  circumscribed 
quadrilateral  is  equal  to  the  sum  of  the  other  two  sides. 

133.  What  is  the  locus  of  the  vertex  of  an  angle  of  con- 
stant magnitude,  whose  sides  pass  through  the  extremities 
of  a  line  of  constant  length  ?  Does  the  locus  consist  of 
one  or  of  two  lines  ? 

134.  If  a  tangent  be  drawn  at  any  point  in  the  convex 
arc  included  between  two  tangents  drawn  from  the  same 
exterior  point,  the  perimeter  of  the  triangle  thus  formed 
is  constant. 

135.  Two  circles  intersect  in  A,  B;  through  B  a  line  is 
drawn  meeting  the  circles  in  C,  2),  respectively;  prove 
that  the  angle  CAD  is  constant  for  all  positions  of  the 
line  CD. 

Hint.  It  is  sufficient  if  the  sum  of  the  angles  C,  D,  is  proved  to 
be  constant. 


THE  CIRCLE  25 

136.  Two  circles  are  tangent  internally  at  A,  and  BC 
is  a  chord  of  the  outer  circle,  tangent  to  the  inner  circle 
at  D;  prove  that  AD  bisects  the  angle  BAG. 

Hint.     Draw  the  common  tangent. 

137.  Find  a  point  equidistant  from  three  lines,  two  of 
which  are  parallel. 

138.  Find  a  point  equidistant  from  two  adjacent  sides 
of  a  quadrilateral,  and  also  equidistant  from  the  other 
two.     Is  there  more  than  one  solution  ? 

139.  The  altitude  of  an  equilateral  triangle  is  9  in.; 
find  the  radius  of  the  inscribed  and  of  the  circumscribed 
circle. 

140.  Any  two  parallel  lines  drawn  through  the  points 
of  intersection  of  two  circles,  and  terminated  by  the  circles, 
are  equal. 

Hint.  From  the  center  of  each  circle,  draw  perpendiculars  to  each 
of  the  chords. 

141.  If  any  side  of  an  inscribed  quadrilateral  be  pro- 
duced, the  exterior  angle  thus  formed  is  equal  to  the 
opposite  interior  angle. 

142.  Divide  a  given  circle  into  two  segments  such  that 
one  of  them  shall  contain  an  angle  twice  as  great  as  the 
other. 

143.  If  two  chords  intersect  and  make  equal  angles 
with  the  line  joining  their  point  of  intersection  to  the 
center,  they  are  equal. 

144.  AB  is  a  fixed  chord  of  a  circle,  and  CD  is  any 
diameter;  prove  that  the  sum  or  difference  of  the  perpen- 
diculars from  (7,  D  to  AB  is  constant. 


26  THE  CIRCLE 

145.  Two  circles  intersect  at  A,  B,  and  through  A  any 
two  secants  are  drawn,  one  terminated  by  the  circles  in  (7, 
2),  and  the  other  in  E,  F,  respectively;  prove  that  the  arcs 
CE,  DF,  subtend  equal  angles  at  B. 

146.  The  bisector  of  any  angle  of  an  inscribed  quadri- 
lateral intersects  the  bisector  of  the  opposite  exterior  angle, 
on  the  circle. 

Is  this  true  of  all  forms  of  inscribed  quadrilaterals  ?  Is 
there  any  kind  of  quadrilateral  that  can  never,  under  any 
circumstances,  become  an  inscribed  quadrilateral  ? 

147.  Circles  described  on  any  two  sides  of  a  triangle  as 
diameters  intersect  on  the  third  side  or  the  third  side 
produced. 

Notice  what  takes  place  when  the  triangle  is  isosceles; 
when  it  is  equilateral. 

148.  Two  circles  intersect  at  J.,  B  ;  through  C,  any 
point  on  one  circle,  lines  CAD,  CBE,  are  drawn  cutting 
the  other  circle  in  D,  E\  prove  that  the  chord  BE  is 
parallel  to  the  tangent  at  (7. 

149.  Two  finite  lines  meet  so  that  the  product  of  the 
segments  of  one  equals  the  product  of  the  segments  of 
the  other  ;  prove  that  the  four  extremities  of  the  two 
lines  are  concyclic. 

150.  ABC  is  a  triangle  right  angled  at  C ;  from  any 
point  B  in  BC  a  perpendicular  BE  is  dropped  on  AB  ; 
prove  that  CBBB  =  AB-  EB. 

151.  Tangents  drawn  to  two  intersecting  circles  from 
any  point  in  their  common  chord  produced,  are  equal. 

152.  Of  all  lines  drawn  through  one  of  the  points  of 
intersection  of  two  circles  and  terminated  by  them,  the 
greatest  is  parallel  to  the  line  of  centers  of  the  two  circles. 


THE  CIRCLE  27 

153.  Find  the  locus  of  the  middle  points  of  lines  drawn 
from  an  external  point  to  a  circle. 

154.  The  quadrilateral  formed  by  tangents  to  a  circle 
drawn  at  the  extremities  of  a  pair  of  diameters  is  a 
rhombus. 

155.  The  bisectors  of  the  angles  of  a  circumscribed 
quadrilateral  are  concurrent. 

156.  The  sides  AD,  BO,  of  the  inscribed  quadrilateral 
ABOB  are  produced  to  meet  at  E,  and  a  circle  is  circum- 
scribed about  ABE ;  prove  that  the  tangent  to  this  circle 
at  E  is  parallel  to  CB. 

157.  Two  circles  are  tangent,  either  internally  or 
externally,  at  A  ;  through  this  point  two  lines  are  drawn 
meeting  one  circle  in  B,  0,  and  the  other  in  2>,  E ;  prove 
that  the  chord  BE  is  parallel  to  BO. 

Hint.     Draw  the  common  tangent. 

158.  If  the  opposite  angles  of  a  quadrilateral  are  supple- 
mentary, the  quadrilateral  is  inscriptible. 

159.  In  a  triangle  if  a  perpendicular  be  drawn  from  one 
extremity  of  the  base  to  the  bisector  of  the  opposite  angle, 

(i)  It  will  make  with  either  of  the  sides  containing  that 
angle  an  angle  equal  to  half  the  sum  of  the  angles  at  the 
base. 

(ii)  It  will  make  with  the  base  an  angle  equal  to  half 
the  difference  of  the  angles  at  the  base. 

160.  The  middle  point  of  an  arc  subtended  by  a  chord 
is  joined  to  the  extremities  of  another  chord  ;  prove  that 
the  triangles  thus  formed  are  similar,  and  the  quadrilateral 
thus  formed  is  inscriptible. 


28  THE  CIRCLE 

161.  Find  in  a  given  line  a  point  such  that  lines  drawn 
from  it  to  two  given  points  are  perpendicular  to  each 
other. 

Note  carefully  the  number  of  solutions. 

162.  The  locus  of  the  vertex  of  a  triangle  having  a 
given  base  and  a  given  angle  at  the  vertex,  is  the  arc 
forming  with  the  base  a  segment  which  will  contain  the 
given  angle. 

163.  The  radii  of  two  concentric  circles  are  a  and  5, 
respectively  ;  find  the  radius  of  a  third  circle  which  will 
be  tangent  to  both  the  concentric  circles,  and  will  contain 
the  smaller. 

164.  A  ladder  rests  with  one  end  on  a  horizontal  floor 
and  the  other  against  a  vertical  wall ;  find  the  locus  of 
the  middle  point  of  the  ladder  as  the  lower  end  slides 
along  the  floor  in  a  direction  at  right  angles  to  the  wall. 

165.  Find  the  locus  of  the  center  of  a  circle  tangent 
to  two  concentric  circles.  Of  how  many  lines  does  the 
locus  consist? 

166.  Six  equal  circles  can  be  drawn  about  a  circle  of 
the  same  radius,  so  that  each  shall  be  tangent  to  the  inner 
circle,  and  to  two  of  the  outer  circles. 

167.  In  the  sides  BC,  CA,  AB,  of  an  equilateral  tri- 
angle ABC  the  points  D,  E,  F,  are  taken  respectively,  so 
that  BD,  CE,  AF,  are  equal,  each  to  each.  The  lines  AB, 
BE,  CF,  are  drawn,  intersecting  in  G,  H,  K:  prove  that 
the  triangle  GHK  is  equilateral. 

168.  Through  one  of  the  points  of  intersection  of  two 
circles  draw  a  chord  of  one  circle  which  shall  be  bisected 
by  the  other. 


THE  CIRCLE  29 

169.  AB,  CD,  are  two  parallel  chords  in  a  circle  ;  prove 
that  the  points  of  intersection  of  AC,  BD,  and  AD,  BO, 

are  collinear  with  the  middle  points  of  the  chords. 

What  other  point  also  lies  in  the  same  line  with  these  ? 

170.  In  a  given  circle  draw  a  chord  equal  to  a  given 
line  and  parallel  to  another  given  line.  Between  what 
limits  in  value  must  the  former  line  lie? 

171.  In  a  diameter  of  a  circle,  produced  indefinitely, 
find  a  point  such  that  the  tangent  drawn  from  it  to  the 
given  circle  shall  be  of  a  given  length. 

Can  this  problem  be  solved  algebraically  ? 

172.  Two  circles  intersect  at  A,  B,  and  through  A  two 
diameters  AC,  AD,  are  drawn,  one  in  each  circle;  prove 
that  the  points  C,  B,  D,  are  collinear. 

173.  Through  a  given  point  without  a  circle,  draw  a 
line  which  shall  cut  off  a  segment  capable  of  containing 
a  given  angle. 

Hint.  Reduce  this  problem  to  the  problem  of  drawing  a  tangent 
to  a  circle  from  a  given  point  without  the  circle. 

174.  Two  intersecting  circles  are  cut  by  a  secant  par- 
allel to  the  common  chord  ;  prove  that  the  two  parts  of  the 
secant  intercepted  between  the  circumferences  are  equal. 

175.  The  sides  of  a  quadrilateral  are  made  the  diameters 
of  circles  ;  prove  that  the  common  chord  of  any  two  con- 
secutive circles  is  parallel  to  the  common  chord  of  the 
other  two. 

Hint.     Prove  by  means  of  No.  41. 

176.  A  circle  having  for  its  center  the  middle  point  of 
one  side  of  a  triangle,  and  a  radius  equal  to  half  the  sum 
of  the  other  two  sides,  is  tangent  to  the  circles  having 
those  sides  as  diameters. 


30  THE  CIRCLE 

177.  The  four  common  tangents  to  two  circles  external 
to  each  other  intersect,  pair  by  pair,  on  the  line  of  centers 
of  the  circles. 

178.  Through  a  point  without  a  circle,  draw  a  secant 
such  that  the  intercepted  chord  shall  have  a  given  value  a. 

179.  Draw  through  a  given  point  a  line  which  shall  be 
equidistant  from  two  other  given  points.    (Two  solutions.) 

180.  Through  a  given  point  A  on  the  circumference  of 
the  outer  of  two  concentric  circles,  draw  a  chord  that 
shall  be  trisected  by  the  inner  circle. 

Hint.  From  A  as  a  center,  with  a  radius  equal  to  the  diameter  of 
the  inner  circle,  strike  an  arc  intersecting  the  inner  circle.  This  point 
of  intersection  will  suggest  a  line  which  will  give  the  solution. 

181.  If  AB  be  a  fixed  chord  of  a  circle,  and  C  any  point 
in  either  of  the  arcs  subtended  by  it,  the  bisector  of  the 
angle  ACB  intersects  the  conjugate  arc  in  the  same  point, 
whatever  be  the  position  of  C. 

182.  AB,  AC,  are  tangents  to  a  circle  from  A,  and  D  is 
any  point  on  the  circumference ;  prove  that  the  sum  of 
the  angles  ABB  and  A  CD  is  constant  for  any  point  on 
the  convex  arc  within  the  angle  A,  but  changes  as  the 
point  D  passes  either  B  or  C 

183.  A,  B,  C,  D,  are  four  points  in  order  on  a  straight 
line,  and  EF  is  a  common  tangent  to  the  circles  having 
AC,  BD,  as  diameters  ;  prove  that  the  angle  CAE  is  equal 
to  the  angle  CEF. 

184.  The  bisectors  of  the  angles  formed  by  producing 
the  opposite  sides  of  an  inscribed  quadrilateral  are  perpen- 
dicular to  each  other. 

Hint.  Prove  by  means  of  the  arcs  which  measure  the  half  angles 
thus  formed. 


THE  CIRCLE  31 

185.  The  locus  of  the  centers  of  circles  inscribed  in  the 
triangles  which  have  a  given  base  and  a  given  angle  at 
the  vertex  is  the  arc  forming  with  the  base  a  segment 
capable  of  containing  an  angle  equal  to  90°  plus  half  the 
given  angle  at  the  vertex. 

186.  Given  any  four  points,  A,  B,  C,  D.  Find  a  point 
E  such  that  each  of  the  angles  AEB,  CED,  shall  equal  a 
right  angle. 

187.  Given  a  chord  AB,  and  a  segment  of  a  circle 
standing  on  it ;  find  the  locus  of  the  intersection  of  the 
medians  of  all  triangles  having  the  chord  as  a  base,  and 
their  vertices  in  the  arc  of  the  segment. 

Hint.  Trisect  the  chord.  The  locus  is  an  arc  on  the  second  of  the 
three  equal  portions. 

188.  Two  parallel  chords  1  in.  apart  are  respectively 
6  in.  and  8  in.  in  length ;  find  the  radius  of  the  circle. 

Is  there  more  than  one  solution? 

189.  A  diameter  of  a  circle  is  indefinitely  produced; 
find  in  it  a  point  such  that  two  tangents  to  the  circle  shall 
contain  a  given  angle. 

190.  On  a  given  line  as  a  base  a  system  of  rhombuses 
is  constructed ;  find  the  locus  of  the  intersection  of  their 
diagonals. 

191.  Through  one  of  the  points  of  intersection  of  two 
circles,  draw  a  line  terminated  by  the  circles  and  bisected 
by  the  point  of  intersection. 

Hint.  From  the  middle  point  of  the  line  of  centers  draw  a  line  to 
the  point  of  intersection  of  the  circles.  The  required  line  can  then 
be  drawn  without  difficulty. 

192.  Through  one  of  the  points  of  intersection  of  two 
circles  a  line  is  drawn  terminated  by  the  circles;  prove 


32  THE  CIRCLE 

that  the  angle  between  the  tangents  at  its  extremities  is 
equal  to  the  angle  between  the  tangents  at  the  point  of 
intersection  of  the  circles. 

193.  Find  a  point  such  that  the  sum  of  the  tangents 
drawn  from  it  to  a  given  circle  shall  equal  the  sum  of  two 
given  lines. 

194.  Draw  a  tangent  to  a  given  circle  so  that  the  part 
contained  within  another  given  circle  shall  equal  a  given 
length  a. 

195.  How  can  the  distance  be  found  between  two 
objects  on  the  ground,  which  are  separated  by  a  lake  ? 

196.  How  can  the  distance  from  an  accessible  to  an 
inaccessible  point  be  found? 

In  connection  with  the  preceding  problem,  the  following  anecdote 
is  related  of  one  of  Napoleon's  engineers.  Corning,  on  one  of  his 
marches,  to  the  bank  of  a  river,  Napoleon  demanded  to  know  the 
width  of  the  stream.  The  engineer  thus  called  upon  said  that  he 
could  not  ascertain  its  width  until  his  instruments  arrived,  which  were 
in  the  rear  with  the  baggage.  The  emperor  insisted  upon  an 
immediate  answer,  but  the  engineer  protested  earnestly  that  it  would 
be  impossible  to  determine  the  distance  without  the  aid  of  the  proper 
instruments.  "Tell  me  without  delay  the  width  of  this  river," 
thundered  the  impatient  emperor.  The  engineer  knew  well  the 
imperious  temper  of  Napoleon,  and  felt  his  heart  sink  as  he  saw 
before  him  disgrace,  and  perhaps  dismissal,  as  the  result  of  failure. 
He  hesitated  but  a  single  instant ;  then,  drawing  himself  up  erect  to 
face  the  stream,  he  pulled  his  cap  down  over  his  eyes  until  the  tip  of 
its  visor  just  touched  the  line  of  sight  to  the  opposite  bank.  Then 
turning  his  body  through  a  right  angle,  still  preserving  his  rigid  atti- 
tude, he  marked  with  his  eye  the  spot  on  the  ground  against  which 
the  tip  of  his  visor  now  rested.  Pacing  off  the  distance  thus  deter- 
mined, he  returned  to  where  the  emperor  was  standing  and  said, 
"Your  Majesty,  the  width  of  the  river  is,  approximately,  so  many 
meters."     The  engineer  was  rewarded  with  instant  promotion. 


THE  CIRCLE  33 

197.  Draw  a  line  parallel  to  a  given  line,  and  meeting 
the  sides  of  an  angle  so  that  the  part  intercepted  between 
the  sides  of  the  angle  shall  equal  a  given  length  a. 

198.  Between  two  points  on  the  same  side  of  a  line, 
find  the  shortest  path  which  shall  touch  the  given  line. 

Hint.     Solve  by  means  of  the  principle  of  symmetry. 

199.  Draw  a  secant  to  two  given  circles  so  that  the 
chords  cut  off  may  have  given  lengths  a,  b. 

200.  Draw  a  tangent  to  a  given  circle  which  shall  be 
(i)  parallel  to  a  given  line;  (ii)  perpendicular  to  a  given 
line. 

201.  Find  the  locus  of  the  centers  of  all  circles  which 
cut  a  given  circle  orthogonally  at  a  given  point. 

202.  A  circle  is  described  on  one  of  the  legs  of  a  right 
triangle  as  diameter;  prove  that  the  tangent  at  the  point 
where  it  cuts  the  hypotenuse  bisects  the  other  leg. 

203.  Two  unequal  segments  of  circles  are  described  on  the 
same  side  of  the  same  chord,  and  the  extremities  of  the  chord 
are  joined  to  any  point  in  the  outer  arc;  prove  that  the 
portion  of  the  inner  arc  between  these  lines  is  constant. 

204.  Two  circles  intersect  at  A,  B,  and  through  O,  any 
point  on  one  of  the  circles,  lines  CA,  CB,  are  drawn  inter- 
secting the  other  circle  in  2),  E,  respectively;  find  the  locus^ 
of  the  intersection  of  AE,  BD. 

Hint.  Prove  that  the  angle  formed  by  the  intersecting  lines  is 
constant. 

205.  A  straight  rod  AB  slides  between  two  rulers 
placed  at  right  angles  to  each  other,  and  from  its  extremi- 
ties AC,  BO,  are  drawn  perpendicular  to  the  respective 
rulers;  find  the  locus  of  0. 

conant's  ex.  geom.  —  3 


34  THE  CIRCLE 

206.  Two  equal  circles  intersect  at  A,  B,  and  through 
A  any  straight  line  CAB  is  drawn  meeting  the  circum- 
ferences in  (7,  2),  respectively;  prove  that  BO  is  equal  to 
BB. 

207.  From  a  point  A,  without  a  circle,  a  secant  ABO 
and  a  tangent  .AZ)  are  drawn ;  if  the  bisector  of  the  angle 
BBC  meets  BCnt.E,  prove  that  AE  =  AB. 

208.  The  bisectors  of  all  angles  inscribed  in  a  given 
segment  are  concurrent. 

Is  the  same  theorem  true  of  the  medians  ? 

209.  Find  the  points  of  intersection  of  the  circles  whose 
diameters  are  the  sides  of  an  equilateral  triangle. 

210.  ABCB  is  an  inscribed  quadrilateral,  and  on  AB 
as  a  chord,  a  circle  is  drawn  cutting  AB,  BC,  in  B,  F,  re- 
spectively; prove  that  EF  is  parallel  to  BC. 

211.  If  a  circle  be  inscribed  in  a  triangle  whose  sides 
are  8,  13,  IT,  respectively,  find  the  value  of  each  of  the 
segments  into  which  the  sides  are  divided  by  the  points 
of  tangency. 

212.  If  a  tangent  be  drawn  to  a  circle  at  the  extremity 
of  a  chord,  the  middle  point  of  the  subtended  arc  is  equi- 
distant from  the  chord  and  the  tangent. 

213.  Two  circles  whose  diameters  are  as  1 :  2  are  tangent 
internally;  prove  that  the  smaller  circle  bisects  every 
chord  of  the  larger  drawn  through  the  point  of  tangency. 

214.  In  a  circle  whose  center  is  0,  the  locus  of  the 
middle  points  of  chords  which  pass  through  a  given  point 
P  is  the  arc,  comprised  within  the  given  circle,  of  a  circle 
having  OP  for  a  diameter. 

Notice  each  of  the  three  cases,  due  to  different  positions 
of  P. 


THE  CIRCLE  35 

215.  If  ABC  be  an  equilateral  triangle,  find  the  locus 
of  a  point  D,  such  that  DA  =  DB  +  DC. 

216.  Find  a  point  such  that  the  three  sides  of  a  triangle 
are  seen  from  it  under  equal  angles. 

When  is  this  impossible  ? 

217.  Construct  a  circle  which  shall  touch  a  given  circle 
in  a  given  point,  and  also  pass  through  another  given 
point. 

218.  C  is  any  point  on  an  arc  whose  chord  is  AB,  and 
the  angles  CAB,  CBA,  are  bisected  by  lines  meeting  at  D ; 
find  the  locus  of  D. 

Is  there  a  corresponding  problem  when  the  exterior 
angles  are  bisected  ? 

219.  The  sides  AB,  B  C,  CD,  of  an  inscribed  quadrilateral 
subtend  arcs  of  95°,  105°,  115°,  respectively ;  BA  and  CD 
produced  meet  at  E,  and  AD,  BC,  at  I7;  find  the  value  in 
degrees  of  the  angle  AFB. 

220.  A  given  line  revolves  on  a  given  point  i  as  a 
pivot ;  find  the  locus  of  the  foot  of  the  perpendicular 
dropped  on  this  line  from  a  second  point  B. 

221.  If  a  circle  be  circumscribed  about  a  triangle,  the 
feet  of  the  perpendiculars  dropped  from  any  point  on  the 
circle  to  the  sides  of  the  triangle  are  collinear. 

Hint.     Prove  by  means  of  inscriptible  quadrilaterals. 

Note.  The  line  connecting  these  points  is  often  called  Simpson's 
line,  from  Robert  Simpson,  professor  of  mathematics  at  the  University 
of  Glasgow,  to  whom  the  discovery  of  this  theorem  is  commonly 
attributed. 

222.  AB  is  a  fixed  diameter  of  a  circle,  and  the  chord 
AC  is  produced  to  D,  so  that  DC=BC;  find  the  locus 
of  D  as  AC  revolves  on  J.  as  a  pivot. 


36  THE  CIRCLE 

223.  The  feet  of  the  medians  and  the  feet  of  the  per- 
pendiculars let  fall  from  the  vertices  of  a  triangle  on  the 
opposite  sides  are  concyclic. 

Hint.  Pass  a  circle  through  the  feet  of  the  medians ;  then  prove 
that  the  foot  of  any  one  of  the  three  perpendiculars  will  lie  on  this 
circle. 

224.  The  bisectors  of  the  angles  of  an  inscribed  triangle 
ABC  meet  the  circle  in  i),  E,  JP,  respectively  ;  find  each 
angle  of  the  triangle  DEF  in  terms  of  the  angles  of  the 
triangle  ABC. 

225.  Two  circles  intersect  at  A,  B,  and  one  of  them 
passes  through  the  center  0  of  the  other ;  prove  that  OA 
bisects  the  angle  between  the  common  chord  and  the  tan- 
gent to  the  first  circle  at  A. 

226.  Two  circles  intersect  at  A,  B,  and  through  any 
point  in  the  chord  AB  two  chords  are  drawn,  one  in  each 
circle ;  prove  that  their  four  extremities  are  concyclic. 

Is  there  any  simple  method  of  finding  a  fourth  propor- 
tional except  the  one  usually  given  in  text-books  on  ele- 
mentary geometry  ? 

227.  AB  is  a  fixed  diameter  of  a  circle  whose  center  is 
0 ;  from  (7,  any  point  on  the  circumference,  a  chord  CD 
is  drawn  perpendicular  to  AB  ;  prove  that  the  bisector  of 
the  angle  OCT)  cuts  the  circle  in  a  fixed  point  as  C  moves 
over  a  semicircle. 

228.  Describe  two  circles  of  given  radii  to  touch  each 
other  and  a  given  line  on  the  same  side  of  the  line. 

How  many  solutions  ? 

229.  A  series  of  circles  touch  a  given  line  at  a  given 
point,  and  are  cut  by  a  line  parallel  to  the  given  line; 
prove  that  all  tangents  to  the  various  circles  at  the  points 


THE  CIRCLE  37 

where  they  are  cut  by  the  second  line  are  tangent  to  a 
fixed  circle  whose  center  is  the  given  point. 

230.  Two  circles  are  tangent  internally,  and  any  third 
circle  is  drawn  touching  both;  prove  that  the  sum  or  the 
difference  of  the  distances  from  the  centers  of  the  two 
given  circles  to  the  center  of  the  third  circle  is  constant. 

231.  Draw  a  line  meeting  the  sides  of  the  angle  A  in 
B,  C,  so  that  BC  equals  a  given  length  a,  and  AB  —  AC 

Two  circles,  of  any  radii  whatever,  are  tangent  exter- 
nally at  A,  and  a  direct  common  tangent  touches  them  at 
B,  C;  if  the  centers  of  the  circles  are  0,  Of,  respectively, 
prove  that  :  — 

232.  The  bisectors  of  the  angles  BOO',  CO'  0,  intersect 
at  right  angles  on  BC 

233.  BC2  is  equal  to  the  product  of  the  diameters  of 
the  given  circles. 

234.  Find  the  shortest  path  between  two  points  lying 
within  the  sides  of  an  angle,  which  shall  touch  both  sides 
of  the  angle. 

Hint.  Prove  by  means  of  points  symmetrical  to  the  respective 
sides. 

235.  Two  circles  intersect,  and  through  each  point  of 
intersection  a  secant  is  drawn  terminated  by  the  circles; 
prove  that  the  chords  joining  their  extremities  are  parallel. 

236.  The  feet  of  the  altitudes  of  a  triangle  are  con- 
nected; prove  that  any  two  sides  of  the  triangle  thus 
formed  make  equal  angles  with  that  side  of  the  original 
triangle  on  which  they  meet. 

237.  Through  the  extremity  B  of  the  diameter  AB  a 
tangent  is  drawn  meeting  the  chords  AC,  AD,  in  U,  F, 


38  THE  CIRCLE 

respectively;   prove  that   the  triangles   A  CD,  AEF,  are 
similar. 

238.  From  any  point  on  a  given  arc  perpendiculars  are 
dropped  upon  radii  drawn  from  the  extremities  of  the  arc; 
prove  that  the  line  joining  the  feet  of  these  perpendiculars 
is  constant. 

239.  ABC and  AB'Cf  are  two  triangles  having  a  com- 
mon angle  A,  and  the  circles  circumscribed  about  them 
meet  at  D;  prove  that  the  feet  of  the  perpendiculars  from 
D  to  AB,  BC,  CA,  B'  C ,  are  collinear. 

Hint.    Prove  by  means  of  Simpson's  line. 

240.  From  a  point  A  without  a  circle  two  tangents  are 
drawn,  and  the  chord  of  contact  is  cut  at  B  by  the  line 
joining  A  to  the  center  ;  prove  that  any  circle  passing 
through  A*  B,  cuts  the  given  circle  orthogonally. 

241.  Divide  a  line  into  two  parts  such  that  their  product 
shall  be  a  maximum. 

242.  If  a  circle  can  be  inscribed  in  an  inscriptible 
quadrilateral,  the  lines  joining  the  opposite  points  of 
contact  are  perpendicular  to  each  other. 

243.  Two  circles  intersect  at  A,  B,  and  at  A  tangents 
are  drawn,  one  to  each  circle,  to  meet  the  circumferences 
at  (7,  2);  prove  that  the  triangles  ABC,  DAB,  are  similar. 

244.  AB  is  a  diameter  of  a  circle,  and  AC,  BD,  are 
chords  intersecting  within  the  circle  at  E ;  prove  that  the 
circle  passing  through  C,  D,  E,  cuts  the  given  circle 
orthogonally. 

245.  The  tangents  drawn  at  the  points  of  contact  of 
three  circles  that  are  tangent  externally,  pair  by  pair,  are 
concurrent  and  equal. 


THE  CIRCLE  39 

246.  A  circle  is  tangent  to  a  given  circle  and  to  a  given 
line;  prove  that  the  two  points  of  tangency  and  the 
extremity  of  the  diameter  of  the  given  circle  perpendicular 
to  the  given  line  are  collinear. 

247.  Draw  a  line  so  that  its  distances  from  two  given 
points  shall  be  a  and  b  respectively. 

248.  Draw  a  line  through  a  given  point  so  that  the 
distances  from  this  point  to  the  feet  of  perpendiculars 
dropped  on  this  line  from  two  other  given  points  shall 
be  equal.     (Two  solutions.) 

249.  Draw  a  diameter  to  a  given  circle  at  a  distance  a 
from  a  given  point  on  the  circle. 

Note.  The  student  should,  in  any  problem  like  the  above,  care- 
fully observe  that  the  construction  is,  in  general,  not  possible.  The 
conditions  under  which  it  is  possible  should  then  be  determined,  and 
the  number  of  constructions  noted. 

250.  Draw  a  chord  to  a  given  circle,  equal  to  a  and  at 
a  distance  b  from  a  given  point  on  the  circle. 

251.  Through  two  circles  exterior  to  each  other  draw  a 
secant  such  that  the  intercepted  chords  shall  be  equal. 

Note.  In  many  constructions  like  the  above,  the  work  can  be 
greatly  simplified  by  reducing  the  problem  to  some  other  problem.  No.  251 
can  readily  be  reduced  to  the  problem  of  drawing  a  tangent  to  two 
given  circles. 

252.  A,  B,  are  the  middle  points  of  the  lesser  arcs  sub- 
tended by  the  chords  CD,  CE,  respectively;  if  the  line 
AB  cut  CD,  CE,  at  F,  a,  respectively,  prove  CF=  CG. 

Is  this  true  for  all  possible  cases? 

253.  The  opposite  sides  of  an  inscribed  quadrilateral 
are  produced  to  meet  in  the  two  exterior  points  A,  B. 
Through   the  point  A  and  the  two  nearest  vertices  of 


40  THE  CIRCLE 

the  quadrilateral  a  circle  is  drawn,  and  through  B  and 
the  two  vertices  of  the  quadrilateral  remote  from  B 
another  circle  is  drawn.  If  these  two  circles  intersect  at 
C,  then  are  the  points  A,  C,  B,  collinear. 

254.  If  three  circles  mutually  intersect  one  another,  the 
common  chords  are  concurrent. 

Hint.  Let  two  of  the  chords  intersect;  from  this  point  of  inter- 
section draw  lines  to  the  two  remaining  points  of  intersection  of  the 
circles;  prove  that  either  of  these  must,  when  produced,  coincide 
with  the  other. 

255.  In  any  triangle  the  altitudes  are  produced  to  meet 
the  circumscribed  circle ;  then  will  each  side  bisect  that 
part  of  the  altitude  perpendicular  to  it  which  lies  between 
the  orthocenter  and  the  circumference. 

256.  If  0  be  the  orthocenter  of  the  triangle  ABC,  the 
angles  BOO,  BAC,  are  supplementary. 

257.  AB  is  a  fixed  chord  of  a  circle,  and  AC,  BD,  are 
any  two  parallel  chords ;  prove  that  the  line  CD  touches 
a  fixed  concentric  circle. 

258.  If  0  be  the  orthocenter  of  the  triangle  ABC,  then 
is  any  one  of  the  four  points  0,  A,  B,  C,  the  orthocenter 
of  the  triangle  whose  vertices  are  the  other  three. 

259.  The  three  circles  which  pass  through  two  vertices 
of  a  triangle  and  its  orthocenter  are  each  equal  to  the 
circle  circumscribed  about  the  triangle. 

260.  On  the  semicircle  whose  diameter  is  AB,  two 
points  C,  D,  are  taken,  and  the  chords  A  C,  BD,  and  AD, 
BC,  are  drawn,  intersecting  (produced  if  necessary)  in 
F,  G-,  respectively;  prove  that  the  line  FQ-  is  perpendicu- 
lar to  AB. 


THE  CIRCLE  41 

261.  ABC  is  an  inscribed  triangle,  0  its  orthocenter, 
and  AK&  diameter ;  prove  that  OCKB  is  a  parallelogram. 

262.  The  line  joining  the  orthocenter  of  an  inscribed 
triangle  to  the  middle  point  of  the  base,  meets  the  circle 
in  the  same  point  as  the  diameter  drawn  from  the  vertex 
of  the  triangle. 

263.  The  perpendicular  from  the  vertex  of  an  inscribed 
triangle  to  the  base,  and  the  line  joining  the  orthocenter 
to  the  middle  point  of  the  base  are  produced  to  meet  the 
circumference  in  ilff,  iV",  respectively ;  prove  that  the  line 
MN  is  parallel  to  the  base. 

264.  The  distance  from  each  vertex  of  a  triangle  to  the 
orthocenter  is  twice  the  perpendicular  from  the  center  of 
the  circumscribed  circle  to  the  opposite  side. 

265.  Three  circles  are  drawn,  each  passing  through  the 
orthocenter  and  two  vertices  of  a  triangle ;  prove  that 
the  triangle  formed  by  joining  their  three  centers  is  simi- 
lar to  the  given  triangle. 

Is  the  triangle  thus  obtained  equal  to  the  given  triangle  ? 

266.  The  bisectors  of  the  angles  of  a  triangle  meet  at 
0,  and  are  produced  to  meet  the  circumscribed  circle  in 
D,  F,  F;  prove  that  0  is  the  orthocenter  of  the  triangle 
DEF. 

267.  The  altitudes  of  the  triangle  ABO  meet  the  sides 
upon  which  they  fall  in  i),  F,  F,  respectively,  and  the 
triangle  DFF  is  drawn ;  prove  that  the  triangles  DFC, 
AFF,  DBF,  are  similar  to  each  other  and  to  the  original 
triangle. 

268.  Given  the  base  and  the  opposite  angle  of  a  tri- 
angle ;  find  the  locus  of  the  orthocenter. 


42  THE  CIRCLE 

269.  Given  the  base  and  the  opposite  angle  of  a  tri- 
angle ;  find  the  locus  of  the  intersection  of  the  angle 
bisectors. 

270.  Two  circles  whose  centers  are  0,  0',  intersect  at 
A,  B,  and  the  line  CAD  is  drawn,  terminated  by  the 
circles;  find  the  locus  of  the  intersection  of  CO,  DO', 
and  prove  that  it  passes  through  B. 

271.  Two  segments  of  circles  are  on  the  same  side  of 
the  common  chord  AB,  and  C,  D,  are  two  points,  one  on 
each  arc ;  find  the  locus  of  the  intersection  of  the  bisectors 
of  the  angles  CAD,  CBD. 

272.  Two  equal  circles  are  tangent  externally  to  each 
other,  and  through  the  point  of  contact  two  chords  are 
drawn,  one  in  each  circle,  at  right  angles  to  each  other ; 
prove  that  the  straight  line  joining  their  other  extremities 
is  equal  to  the  diameter  of  either  circle. 

273.  Given  three  points  not  in  the  same  straight  line  ; 
show  how  to  find  any  number  of  points  in  the  circum- 
ference of  the  circle  passing  through  them,  without  find- 
ing the  center. 

274.  Two  equal  circles  intersect  at  A,  B,  and  a  third 
circle,  whose  center  is  A,  and  whose  radius  is  less  than 
AB,  intersects  them  on  the  same  side  of  AB  in  the  points 
C,  D  ;  prove  that  the  points  C,  D,  B,  are  collinear. 

275.  Two  circles  intersect  at  A,  B,  and  through  A  two 
lines,  CAD,  EAF,  are  drawn,  terminated  by  the  circles;  if 
CE,  DF,  intersect  at  G,  prove  that  C,  B,  D,  Q-,  are  con- 
cyclic. 

276.  On  the  same  side  of  the  same  chord  three  seg- 
ments of  circles  are  constructed,  containing,  respectively, 
a  given  angle,  its  supplement,  and  a  right  angle;  prove 


THE  CIRCLE  43 

that  the  intercept  made  by  the  two  former  arcs  on  any 
secant  drawn  through  either  extremity  of  the  given  chord 
is  bisected  by  the  last-named  arc. 

277.  On  the  sides  BO,  OA,  AB,  in  order,  of  the  triangle 
ABO,  any  points  D,  E,  F,  are  taken;  prove  that  the  circles 
circumscribed  about  the  triangles  AEF,  BFD,  CUD,  are 
concurrent. 

278.  ABO  is  an  inscribed  triangle,  and  from  any  point 
on  the  circle  perpendiculars  are  drawn  to  BO,  OA,  AB, 
respectively,  meeting  the  circle  in  A1 ',  B' ',  0' ;  prove  that 
the  triangle  A'B'  0'  is  equal  to  ABO. 

279.  Two  tangents  are  drawn  from  a  point  to  a  circle; 
prove  that  the  square  of  the  radius  of  the  circle  is  equal 
to  the  product  of  the  line  joining  the  external  point  and 
the  center,  multiplied  by  the  inner  segment  of  that  line 
made  by  the  chord  of  contact. 

280.  From  the  vertices  A,  B,  of  the  triangle  ABO,  per- 
pendiculars AD,  BE,  are  dropped  on  the  opposite  sides 
respectively;  prove  A O  •  OE=BO-  OB. 

281.  From  any  point  in  the  common  chord,  produced,  of 
a  system  of  circles  passing  through  two  fixed  points,  tan- 
gents are  drawn  to  all  the  circles;  prove  that  the  locus  of 
the  points  of  tangency  is  a  circle  cutting  all  the  given 
circles  orthogonally. 

282.  AB  is  a  fixed  diameter  of  a  circle,  and  OD  is  a  line 
perpendicular  to  AB,  produced  if  necessary;  if  any  line 
through  A  cuts  OD  at  E,  and  the  circle  at  F,  then  is  the 
product  AE  •  AF  constant. 

283.  Find  at  what  point  in  a  given  line  the  angle  sub- 
tended by  the  line  joining  two  given  points  on  the  same 
side  of  the  given  line  is  a  maximum. 


44  THE  CIRCLE 

284.  Given  two  lines  AB,  AC,  and  I  point  B  between 
them;  prove  that,  of  all  lines  through  B  terminated  by 
the  two  given  lines,  that  which  is  bisected  at  B  cuts  off 
the  minimum  triangle. 

Hint.  Draw  through  D  the  line  which  is  bisected  at  that  point, 
and  draw  through  D  any  other  line  terminated  by  AB,  AC,  in  E,  F, 
respectively.  From  one  of  the  points  E,  F,  draw  a  line  lying  within 
the  sides  of  the  angle  and  terminated  by  the  line  first  drawn.  Two 
equal  triangles  are  now  available,  by  the  aid  of  which  the  theorem 
can  be  proved. 

285.  Find  a  point  on  a  circle  such  that  the  sum  of  the 
squares  of  the  distances. to  this  point  from  two  fixed 
points  on  the  circle  shall  be  a  minimum. 

Hint.  Prove  by  means  of  the  theorem  which  states  that  in  any 
triangle  the  sum  of  the  squares  on  two  sides  is  equal  to  twice  the 
square  on  half  the  third  side  plus  twice  the  square  on  the  median 
drawn  to  the  third  side. 

286.  Given  the  base  and  the  opposite  angle  of  a  triangle; 
find  the  locus  of  the  centroid. 

287.  Given  the  base  and  the  opposite  angle  of  a  triangle; 
find  the  locus  of  the  intersection  of  the  bisectors  of  the 
exterior  base  angles. 

288.  Find  the  locus  of  the  intersection  of  two  lines 
intercepting  on  a  circle  an  arc  of  constant  length,  and 
also  passing  through  two  fixed  points  on  the  circle. 

289.  AB  is  any  diameter  of  a  circle,  and  C,  D,  are  two 
fixed  points  on  the  circle;  find  the  locus  of  the  intersec- 
tion of  AC,  BB. 

290.  Of  all  triangles  that  can  be  inscribed  in  a  given 
triangle,  that  formed  by  joining  the  feet  of  the  altitudes 
has  the  minimum  perimeter. 


THE  CIRCLE  45 

291.  Two  circles  whose  centers  are  0,  0' ,  intersect  at 
A,  B,  and  any  line  CAD  is  drawn,  terminated  by  the 
circles;  prove  that  the  angle  CBD  is  equal  to  the  angle 
OAO'. 

292.  Three  equal  circles  intersect  at  the  point  A,  and 
their  other  points  of  intersection  are  B,  C,  D  ;  prove  that 
each  of  these  four  points  is  the  orthocenter  of  the  triangle 
formed  by  joining  the  other  three. 

293.  From  a  given  point  without  a  circle,  draw  a  line 
to  the  concave  arc  of  the  circle  which  shall  be  bisected  by 
the  convex  arc. 

294.  Draw  a  line  cutting  two  concentric  circles  so  that 
the  chord  intercepted  by  the  greater  shall  be  twice  the 
chord  intercepted  by  the  smaller  circle. 

295.  From  any  point  A  on  the  circumference  of  a  circle 
a  perpendicular  AB  is  drawn  to  (72),  any  chord  of  the 
same  circle,  and  CE  is  drawn  perpendicular  to  the  tangent 
at  A  ;  prove  that  BE  is  parallel  to  DA, 

296.  ABC  is  an  inscribed  triangle,  and  from  D,  any 
point  on  the  circle,  perpendiculars  are  drawn  to  the  sides 
BC,  CA,  AB,  meeting  the  circle  in  A ',  B1 ',  <?',  respectively ; 
prove  that  AA',  BBf,  CC ',  are  parallel. 

297.  From  any  point  in  AB,  the  common  chord  of  two 
equal  circles,  a  perpendicular  is  drawn  to  meet  the  circles 
on  the  same  side  of  AB  in  C,  D ;  prove  that  CD  is  of 
constant  length. 

298.  From  any  point  A  on  the  circumference  of  a  circle 
a  perpendicular  AB  is  drawn  to  CD,  any  diameter  of  the 
same  circle,  and  on  CB,  BD,  as  diameters  circles  are 
described  which  cut  CA,  AD,  at  E,  F,  respectively  ;  prove 
that  EF  is  the  common  tangent  to  these  two  circles. 


46  THE  CIRCLE 

299.  Two  straight  lines  of  indefinite  length  are  tangent 
to  a  given  circle,  and  any  chord  is  drawn  so  as  to  be 
bisected  by  the  chord  of  contact ;  if  the  former  chord  be 
produced,  prove  that  the  intercepts  on  it  between  the  cir- 
cumference and  the  tangents  are  equal. 

300.  D  is  any  point  on  the  circumscribed  circle  of  the 
triangle  ABC;  prove  that  the  angle  between  BC  and 
Simpson's  line  for  the  point  D  is  equal  to  the  angle 
between  AD  and  the  diameter  of  the  circle  passing 
through  A.     (See  No.  221.) 

Hint.    Prove  by  means  of  inscriptible  quadrilaterals. 


III.     CONSTRUCTIONS 

Construct  a  circle  with  its  center  in  a  given  line  which 
shall  be :  — 

301.  Tangent  at  a  given  point  to  another  given  line. 

302.  Tangent  to  two  other  given  lines. 

303.  Tangent  at  a  given  point  to  a  given  circle. 
When  is   the  center  of   the  required  circle  infinitely 

distant  ? 

Construct  a  circle  with  its  center  on  a  given  circle  which 
shall  be :  — 

304.  Tangent  to  two  given  lines. 

305.  Tangent  to  a  given  circle  at  a  given  point. 

306.  Tangent  to  a  given  line  at  a  given  point. 

307.  Construct  a  circle  which  shall  be  tangent  to  a 
given  line,  and  to  a  given  circle  at  a  given  point. 

308.  Construct  a  circle  which  shall  be  tangent  to  a 
given  circle,  and  to  a  given  line  at  a  given  point. 

309.  Construct  a  triangle,  having  given  the  vertex,  the 
orthocenter,  and  the  center  of  the  circumscribed  circle. 

Hint.  Bisect  that  part  of  the  altitude  drawn  from  the  given 
vertex,  which  lies  between  the  orthocenter  and  the  circumference  of 
the  circumscribed  circle. 

Construct  a  circle  of  given  radius  r,  which  shall:  — 

310.  Pass  through  a  given  point  and  be  tangent  to  a 
given  line. 

47 


48  CONSTRUCTIONS 

311.  Pass  through  a  given  point  and  be  tangent  to  a 
given  circle. 

312.  Pass  through  a  given  point  and  cut  a  given  circle 
in  a  diameter. 

313.  Pass  through  a  given  point  and  cut  a  given  circle 
orthogonally. 

Hint.  Remember  that  at  any  given  point  on  the  circle  the 
direction  of  the  circle  and  the  direction  of  the  tangent  at  that  point 
are  the  same. 

314.  Be  tangent  to  a  given  line  and  to  a  given  circle. 

315.  Be  tangent  to  two  given  circles. 

316.  Be  tangent  to  a  given  circle  and  cut  another  given 
circle  orthogonally. 

317.  Be  tangent  to  a  given  line  and  cut  a  given  circle 
orthogonally. 

318.  Be  tangent  to  a  given  line  and  cut  a  given  circle 
in  a  diameter. 

319.  Cut  from  two  given  lines  chords  having  given 
lengths  a  and  b. 

Note.  Many  problems  like  the  last  ten  can  be  most  readily  solved 
by  what  is  known  as  the  "  method  of  loci."  Give  a  brief  description 
of  this  method. 

320.  Construct  a  triangle  having  given  the  middle 
points  of  the  three  sides. 

321.  Construct  a  circle  with  its  center  in  a  given  line, 
which  shall  cut  this  line  at  a  given  point  and  be  tangent 
to  another  given  line. 

When  is  the  number  of  solutions  infinite  ? 

322.  Construct  a  circle  with  its  center  in  a  given  line, 


CONSTRUCTIONS  49 

which  shall  cut  this  line  at  a  given  point  P  and  be  tangent 
to  a  given  circle. 

Hint.  Take  PA  on  the  given  line  eqnal  to  the  radius  of  the  given 
circle,  and  join  the  point  A  with  the  center  of  the  circle. 

323.  Construct  a  circle  which  shall  pass  through  a 
given  point  and  be  tangent  to  a  given  circle  at  a  given 
point. 

324.  Construct  a  circle  which  shall  be  tangent  to  three 
given  lines. 

How  many  solutions  are  there  ? 

325.  Construct  a  circle  which  shall  be  tangent  to  two 
given  lines,  one  of  them  at  a  given  point. 

326.  The  chord  of  a  given  segment  is  produced  to  a 
given  length  ;  on  the  part  produced,  construct  a  segment 
similar  to  the  given  segment. 

Two  segments  are  said  to  be  similar  when  their  arcs 
contain  the  same  number  of  degrees,  each  to  each. 

327.  Given  a  circle  and  its  center  ;  find  the  side  of 
the  inscribed  square  by  means  of  the  compasses  alone. 

Note.  This  problem  is  sometimes  called  "  Napoleon's  problem." 
It  is  said  to  have  been  proposed,  and  also  to  have  been  solved,  by  the 
emperor,  Napoleon  Bonaparte,  who  was  especially  fond  of  mathe- 
matics. 

Through  a  given  point  P,  draw  a  line  meeting  the  sides 
of  an  angle  A,  in  the  points  B,  (7,  so  that :  — 

328.  AB  and  A  0  are  equal. 

329.  PB  and  P(7are  equal. 

330.  BC  is  equal  to  twice  AC 

331.  BO  is  equal  to  B A. 

conant's  ex.  geom. — 4 


50  CONSTRUCTIONS 

Draw  a  line  meeting  the  sides  CA,  CB,  of  the  triangle 
ABC  in  D,  E,  respectively,  so  that:  — 

332.  DE  is  parallel  to  AB,  and  equal  to  BE. 
Hint.    Bisect  the  angle  B. 

333.  DE  is  parallel  to  AB,  and  CD  is  equal  to  BE. 

334.  DE  is  equal  to  a  given  length  a,  and  CD  is  equal 
to  BE. 

Hint.  Suppose  the  problem  solved ;  then  through  C,  E,  draw  lines 
parallel  to  DE,  A  C,  respectively. 

335.  DE  is  parallel  to  AB,  and  DE  is  equal  to  the  sum 
of  AD  and  BE. 

336.  DE  is  equal  to  CD,  and  CD  is  equal  to  BE. 
Hint.     Suppose  the  problem  solved ;  then  draw  BD. 

337.  Construct  an  equilateral  triangle  such  that  its 
sides  shall  pass  through  three  given  points. 

338.  Construct  a  triangle  similar  to  a  given  triangle, 
with  its  sides  passing  through  three  given  points. 

339.  Given  the  sum  of  two  lines,  and  a  square  equal  in 
area  to  the  rectangle  whose  adjacent  sides  are  the  two 
lines  respectively.     Find  the  two  lines. 

340.  Given  the  sum  of  two  adjacent  sides  of  a  rec- 
tangle ;  to  inscribe  the  rectangle  in  a  given  circle. 

341.  Given  the  sum  of  two  lines,  and  the  sum  of  the 
squares  on  them  ;  determine  each  of  the  lines. 

Hint.  Let  the  required  lines  CA,  CB,  be  the  legs  of  a  right 
triangle. 

Construct  a  triangle,  given :  — 

342.  One  angle,  the  side  opposite,  and  the  altitude  on 
that  side. 

Is  there  more  than  one  solution  ? 


CONSTRUCTIONS  51 

343.  One  angle,  and  the  two  segments  into  which  the 
opposite  side  is  divided  by  the  bisector  of  that  angle. 

344.  The  radius  of  the  circumscribed  circle,  and  the 
segments  into  which  one  of  the  sides  is  divided  by  the 
bisector  of  the  opposite  angle. 

345.  One  angle,  the  side  opposite,  and  the  angle  between 
the  median  from  the  given  angle  and  one  of  the  adjacent 
sides. 

346.  A  side,  the  median  on  that  side,  and  the  altitude 
on  one  of  the  remaining  sides. 

347.  One  of  the  angles,  a  right  angle,  the  altitude  on 
the  hypotenuse,  and  one  of  the  acute  angles. 

348.  The  triangle  isosceles,  the  altitude  on  the  base, 
and  one  of  the  base  angles. 

349.  Two  sides,  and  the  altitude  on  the  third  side. 

350.  Two  angles,  and  the  altitude  drawn  from  one  of 
them. 

351.  Two  sides,  and  the  altitude  on  one  of  them. 

352.  The  base,  one  of  the  base  angles,  and  one  of  the 
angles  between  the  base  and  the  median  on  the  base. 

353.  The  base,  the  altitude  on  the  base,  and  one  of  the 
angles  between  the  base  and  the  median  on  the  base. 

354.  Two  angles,  and  the  bisector  of  one  of  them. 

355.  An  angle,  its  bisector,  and  one  side  adjacent  to  the 
given  angle. 

356.  An  angle,  its  bisector,  and  one  of  the  segments 
into  which  the  given  bisector  divides  the  side  on  which  it 
falls. 


52  CONSTRUCTIONS 

357.  An  angle,  its  bisector,  and  the  altitude  let  fall 
from  the  vertex  of  this  angle. 

358.  An  altitude  and  an  angle  bisector  drawn  from  the 
same  vertex,  and  one  of  the  segments  into  which  the 
given  bisector  divides  the  side  to  which  it  is  drawn. 

359.  A  side,  the  altitude  on  one  of  the  other  sides,  and 
one  of  the  segments  into  which  the  last-named  side  is 
divided  by  the  bisector  of  the  opposite  angle. 

360.  The  difference  of  the  two  base  angles,  one  of  the 
segments  into  which  the  base  is  divided  by  the  altitude  on 
it,  and  the  corresponding  segment  of  the  base  made  by 
the  bisector  of  the  opposite  angle. 

Hint.  Using  the  outer  extremity  of  the  given  segment  as  a  vertex, 
construct  on  the  inner  side  of  the  segment  an  angle  equal  to  half  the 
given  difference  of  the  base  angles. 

361.  A  side,  the  angle  bisector  drawn  to  one  of  the  other 
sides,  and  the  difference  of  the  segments  of  the  last- 
named  side,  which  are  formed  by  the  altitude  and  the 
angle  bisector,  as  described  in  No.  360. 

362.  Inscribe  a  rectangle  in  a  circle,  having  given  the 
difference  of  two  adjacent  sides  of  the  rectangle. 

Hint.  Consider  the  problem  solved.  Take  on  the  longer  of  the 
two  sides  of  the  rectangle  a  length  equal  to  the  shorter  side,  and 
complete  the  isosceles  triangle.  A  diagonal  and  the  given  difference 
now  form  two  sides  of  a  triangle,  which  can  be  constructed. 

363.  Construct  three  equal  circles  such  that  each  shall 
be  tangent  externally  to  the  other  two,  and  to  a  given 

circle. 

364.  Construct  three  equal  circles  such  that  each  shall 
be  tangent  externally  to  the  other  two,  and  internally  to 
a  given  circle. 


CONSTRUCTIONS  53 

365.  In  a  square  ABCD,  construct  an  equilateral  tri- 
angle AEF  so  that  E,  F,  shall  lie  on  sides  of  the  square. 

Construct  a  triangle,  given :  — 

366.  Two  sides,  and  the  median  on  the  third  side. 

Hint.  Consider  the  problem  solved.  Then  complete  the  parallelo- 
gram. 

367.  The  base  angles,  and  the  median  on  the  included 
side. 

368.  An  angle,  and  the  median  and  the  altitude  drawn 
from  the  vertex  of  the  given  angle  to  the  opposite  side. 

Hint.  In  the  completed  parallelogram,  note  the  relation  which 
exists  between  the  given  angle  and  the  angles  adjacent  to  it. 

369.  The  median  on  one  of  the  sides,  and  the  angles 
made  by  the  given  median  with  each  of  the  other  sides. 

370.  An  angle,  the  altitude  on  the  side  opposite,  and 
the  median  on  one  of  the  remaining  sides. 

371.  Construct  three  equal  circles  tangent  to  each 
other,  pair  by  pair,  so  that  the  area  inclosed  within  the 
convex  arcs  shall  be  2  sq.  in. 

372.  Construct  three  equal  circles  in  an  equilateral  tri- 
angle so  that  each  shall  be  tangent  to  the  other  two  and 
to  two  sides  of  the  triangle. 

373.  Construct  in  a  square  four  equal  circles  so  that 
each  shall  be  tangent  to  two  others  and  to  two  sides  of 
the  square. 

374.  Construct  in  a  given  triangle  a  semicircle  having 
its  diameter  on  one  side,  and  having  the  other  two  sides 
as  tangents. 

375.  Inscribe  a  circle  in  a  given  sector. 


54  CONSTRUCTIONS 

376.  Construct  a  right  triangle,  given  one  of  the  acute 
angles  and  the  sum  of  the  hypotenuse  and  the  side  ad- 
jacent to  the  given  angle. 

Construct  a  triangle,  given  :  — 

377.  One  of  the  base  angles,  the  altitude  on  the  base, 
and  the  sum  of  the  other  two  sides. 

378.  The  base,  the  difference  between  the  base  angles, 
and  the  sum  of  the  other  two  sides. 

Hint.  Suppose  the  problem  solved.  With  the  vertex  as  a  center 
and  the  shorter  of  the  two  sides  as  a  radius,  describe  a  circle  cutting 
the  longer  side  (produced)  in  two  points,  and  join  these  two  points 
to  the  intersection  of  the  base  and  the  shorter  side.  This  forms  an 
auxiliary  triangle,  for  whose  construction  sufficient  data  are  given. 

379.  One  of  the  angles  a  right  angle,  one  of  the  sides 
about  the  right  angle,  and  the  difference  between  the 
hypotenuse  and  the  remaining  side. 

Hint.  Let  ABC  be  the  required  triangle,  right  angled  at  C,  and 
BC  the  given  side.  Produce  AC  to  E,  making  AE  equal  to  AB, 
and  draw  BE. 

380.  The  triangle  isosceles,  the  base,  and  the  difference 
between  one  of  the  legs  and  the  altitude  on  the  base. 

381.  The  base,  the  smaller  of  the  base  angles,  and  the 
difference  between  the  two  remaining  sides. 

382.  An  angle,  the  difference  between  the  adjacent 
sides,   and   the   difference   between   the    two    remaining 

angles. 

Hint.  In  the  constructed  triangle,  measure  off  from  the  given 
vertex  on  the  longer  side  the  length  of  the  shorter,  and  join  the  inter- 
section thus  formed  to  the  extremity  of  the  shorter  side.  From  the 
auxiliary  triangle  thus  formed  at  the  base,  the  required  triangle  can 
be  constructed. 


CONSTRUCTIONS  55 

383.  The  altitude  on  the  base,  the  greater  of  the  base 
angles,  and  the  difference  between  the  two  remaining 
sides. 

384.  The  perimeter,  the  altitude  on  the  base,  and  the 
angle  opposite  the  base. 

Hint.  Produce  the  base  of  the  constructed  triangle  at  each  ex- 
tremity by  a  length  equal  to  the  adjacent  side,  and  join  the  extremi- 
ties of  the  extended  base  to  the  vertex. 

385.  The  perimeter,  the  greater  of  the  base  angles,  and 
the  shorter  of  the  segments  into  which  the  base  is  divided 
by  the  altitude  on  the  base. 

386.  The  two  base  angles,  and  the  difference  between 
the  sum  of  the  two  shorter  sides  and  the  longest  side  —  in 
this  case  the  base. 

Hint.  Through  the  vertex  C  of  the  constructed  triangle  ABC 
produce  one  of  the  shorter  sides,  BC,  to  D  making  CD  =  AC.  On 
BD  mark  off  BE  =  AB,  connect  AD  and  AE.  An  auxiliary  triangle 
ABE  is  thus  formed,  which  can  be  constructed  from  the  given  data. 

387.  The  segments  into  which  the  base  is  divided  by 
the  bisector  of  the  opposite  angle,  and  either  of  the  base 
angles. 

388.  The  two  base  angles,  and  the  greater  of  the  seg- 
ments into  which  the  base  is  divided  by  the  bisector  of 
the  opposite  angle. 

389.  In  a  given  square  construct  a  square  having  a 
given  side,  so  that  its  vertices  shall  lie  on  the  sides  of  the 
given  square. 

Hint.  The  squares  have  the  same  center.  Use  the  semi-diago- 
nals. 

390.  Inscribe  a  square  in  the  part  common  to  two  equal 
intersecting  circles. 


56  CONSTRUCTIONS 

Construct  a  triangle,  given :  — 

391.  The  three  medians. 

Hint.  With  double  the  medians  as  sides,  construct  a  triangle. 
Complete  the  parallelogram,  and  draw  the  other  diagonal. 

392.  Two  sides,  and  the  difference  between  the  angles 
opposite  those  sides. 

Hint.  In  the  required  triangle  A B C  let  AC,  BC,  be  the  given 
sides.  Draw  CD  equal  to  CA  to  meet  the  base  in  D.  What  is  the 
size  of  the  angle  BCD1 

393.  Two  sides,  and  the  difference  between  the  seg- 
ments into  which  the  third  side  is  divided  by  the  altitude 
on  it. 

394.  The  difference  between  the  base  angles,  the  greater 
of  the  two  sides  lying  opposite  the  base  angles,  and  the 
difference  between  the  segments  into  which  the  base  is 
divided  by  the  altitude  on  it. 

395.  The  smaller  of  the  base  angles,  the  difference  be- 
tween the  two  remaining  sides,  and  the  difference  between 
the  segments  of  the  base  made  by  the  altitude  on  it. 

396.  One  angle  a  right  angle,  the  difference  between 
the  legs,  and  the  difference  between  the  segments  of  the 
hypotenuse  made  by  the  altitude  on  it. 

Hint.  In  the  completed  triangle  ABC  let  C  be  the  right  angle, 
and  BC>  AC.  Draw  the  circle  suggested  in  the  hint  to  No.  378,  and  let 
it  cut  BC  in  E,  BC  produced  in  D,  and  AB  in  F.  The  triangle  BFE 
forms  an  auxiliary  triangle,  which  can  be  constructed  if  the  value  of 
the  angle  BFE  can  be  ascertained. 

397  The  smaller  of  the  base  angles,  and  the  segments 
into  which  the  base  is  divided  by  the  bisector  of  the 
opposite  angle. 


CONSTRUCTIONS  57 

398.  The  difference  between  the  base  angles,  and  the 
segments  of  the  base  as  in  the  preceding  problem. 

399.  The  shorter  of  the  segments  of  the  base  made  by 
the  bisector  of  the  opposite  angle,  the  difference  between 
the  base  angles,  and  the  difference  between  the  two 
remaining  sides. 

400.  The  smaller  of  the  base  angles,  the  angle  opposite 
the  base,  and  the  difference  between  the  segments  of  the 
base  as  in  No.  397. 

401.  The  radius  of  the  circumscribed  circle,  the  triangle 
isosceles,  and  one  of  the  base  angles. 

402.  The  base,  the  larger  of  the  base  angles,  and  the 
radius  of  the  circumscribed  circle. 

403.  The  base  angles  and  the  radius  of  the  circum- 
scribed circle. 

404.  The  larger  of  the  base  angles,  the  difference 
between  the  other  two  sides,  and  the  radius  of  the 
circumscribed  circle. 

405.  The  angle  opposite  the  base,  the  sum  of  the  sides 
adjacent  to  this  angle,  and  the  radius  of  the  circumscribed 
circle. 

406.  The  segments  of  the  base  made  by  the  bisector  of 
the  opposite  angle,  and  the  radius  of  the  circumscribed 
circle. 

407.  The  median  on  the  base,  the  difference  between 
the  segments  into  which  the  base  is  divided  by  the  alti- 
tude on  it,  and  the  radius  of  the  circumscribed  circle. 

Hint.  The  given  difference  may  be  conveniently  represented  by  a 
chord  parallel  to  the  base  of  the  triangle. 


58  CONSTRUCTIONS 

408.  The  altitude  and  the  median  on  the  base,  and  the 
radius  of  the  circumscribed  circle. 

409.  The  base  angles,  and  the  radius  of  the  inscribed 
circle. 

410.  The  base,  the  greater  of  the  base  angles,  and  the 
radius  of  the  inscribed  circle. 

411.  The  altitude  on  the  base,  either  of  the  base  angles, 
and  the  radius  of  the  inscribed  circle. 

412.  The  altitude  on  the  base,  the  greater  of  the  seg- 
ments into  which  that  altitude  divides  the  base,  and  the 
radius  of  the  inscribed  circle. 

413.  One  of  the  angles,  its  bisector,  and  the  radius  of 
the  inscribed  circle. 

Construct  a  rectangle,  given :  — 

414.  A  diagonal,  and  the  sum  of  two  adjacent  sides. 

415.  The  sum  of  two  adjacent  sides,  and  the  angle 
between  a  side  and  a  diagonal. 

416.  The  difference  between  a  diagonal  and  one  side, 
and  the  angle  between  a  diagonal  and  the  other  side. 
(See  No.  362.) 

Construct  a  rhombus,  given :  — 

417.  A  side,  and  the  sum  of  the  diagonals. 

418.  An  angle,  and  the  sum  of  the  diagonals. 

419.  An  angle,  and  the  sum  of  a  side  and  the  longer 
diagonal. 

Construct  a  rhomboid,  given :  — 

420.  The  altitude,  a  diagonal,  and  the  angle  between 
the  diagonals. 

421.  An  angle,  and  the  diagonals. 


CONSTRUCTIONS  59 

422.  A  side,  the  difference  between  the  diagonals,  and 
the  angle  between  the  diagonals.  ^ 

423.  An  angle,  a  diagonal,  and  the  sum  of  two  adjacent 
sides. 

424.  An  angle,  the  longer  diagonal,  and  the  difference 
between  one  of  the  shorter  sides  and  the  altitude. 

425.  An  angle,  one  of  the  shorter  sides,  and  the  difference 
between  the  longer  diagonal  and  one  of  the  longer  sides. 

Construct  an  isosceles  trapezoid,  given :  — 

426.  A  diagonal,  a  leg,  and  the  altitude. 

427.  A  base  angle,  a  diagonal,  and  the  sum  of  the 
longer  base  and  a  leg. 

Construct  a  trapezoid,  given  :  — 

428.  The  altitude,  one  of  the  base  angles,  the  greater 
base,  and  the  side  adjacent  to  the  greater  base,  remote 
from  the  given  angle. 

429.  The  greater  base,  the  diagonals,  and  the  angle 
between  the  diagonals. 

Note.  In  constructing  trapezoids,  some  or  all  the  following  lines 
will  be  found  useful.  Let  A  BCD  be  the  trapezoid,  and  AB  the 
greater  base.  Draw  CE  parallel  to  DB  to  meet  AB  produced;  CF 
parallel  to  DA  ;  AH,  EH,  parallel  to  CE,  CA,  respectively;  and  HF, 
HB,  HC. 

430.  The  bases,  one  of  the  remaining  sides,  and  the 
angle  between  the  diagonals. 

431.  The  altitude,  the  base  angles,  and  the*  angle  be- 
tween the  diagonals. 

Hint.     See  note  to  No.  329.     What  is  the  size  of  the  angle  CAH1 

432.  The  sum  of  the  angles  at  the  base,  the  legs,  and 
the  angle  between  the  diagonals. 


60  CONSTRUCTIONS 

433.  The  difference  between  the  bases,  the  other  two 
sides,  and  the  angle  between  the  diagonals. 

434.  The  base  angles,  the  altitude,  and  the  difference 
between  the  greater  base  and  one  of  the  adjacent  sides. 

435.  The  sum  of  the  greater  base  and  one  of  the  adja- 
cent sides,  the  other  two  sides,  and  the  angle  between  the 
two  sides  whose  sum  is  given.     How  many  constructions  ? 

436.  The  sum  of  the  bases,  the  other  two  sides,  and  one 
of  the  angles  at  the  base. 

Hint.     Find  the  difference  between  the  bases. 

437.  The  diagonals,  the  angle  between  them,  and  the 
difference  between  the  greater  base  and  one  of  the  adja- 
cent sides.     (See  Nos.  429  and  362.) 

Construct  an  inscribed  quadrilateral,  given :  — 

438.  The  radius  of  the  circumscribed  circle,  two  adja- 
cent sides,  and  the  angle  between  the  diagonals. 

439.  The  radius  of  the  circumscribed  circle,  one  of  the 
diagonals,  the  angle  at  the  vertex  from  which  that  diago- 
nal is  drawn,  and  the  angle  between  the  diagonals. 

440.  The  radius  of  the  circumscribed  circle,  the  differ- 
ence between  two  adjacent  sides,  the  diagonals  joining 
the  extremities  of  these  sides,  and  the  angle  between  the 
diagonals. 

441.  A  side,  the  two  adjacent  angles,  and  a  diagonal. 

442.  A  side,  two  angles  adjacent  to  each  other  and  one 
of  them  adjacent  to  the  given  side,  and  the  angle  between 
the  diagonals. 

Hint.  The  diagonals  divide  the  quadrilateral  into  two  pairs  of 
similar  triangles.  Find  the  angle  formed  by  a  diagonal  and  one  of 
the  sides  adjacent  to  the  given  side. 


CONSTRUCTIONS  61 

443.  The  sum  of  a  diagonal  rf  and  a  side  a,  the  side 
opposite  a,  and  a  third  side  meeting  a  in  the  vertex 
remote  from  the  intersection  of  a  and  d. 

444.  The  diagonals,  an  angle,  and  one  of  the  angles 
between  the  diagonals. 

Construct  a  tangent  trapezoid,  given :  — 

445.  The  radius  of  the  inscribed  circle,  one  leg,  and  an 
angle  adjacent  to  the  other  leg. 

446.  The  bases,  and  one  of  the  legs. 

Hint.  The  sum  of  two  opposite  sides  of  a  tangent  quadrilateral 
is  equal  to  the  sum  of  the  other  two  sides. 

447.  The  radius  of  the  inscribed  circle,  the  shorter 
base,  and  the  difference  between  the  longer  base  and  a 
leg. 

448.  The  sum  of  the  bases,  and  the  base  angles. 
Construct  a  tangent  quadrilateral,  given :  — 

449.  Two  adjacent  angles,  a  diagonal,  and  the  radius  of 
the  inscribed  circle. 

450.  Two  opposite  sides,  the  sum  of  the  angles  adja- 
cent to  one  of  them,  and  the  radius  of  the  inscribed 
circle. 

451.  Two  adjacent  sides,  and  the  angles  adjacent  to 
one  of  them. 

452.  Two  adjacent  angles,  a  side  adjacent  to  one  of 
them,  and  the  diagonal  drawn  from  the  other. 

453.  Three  sides  and  one  of  the  angles  included  by  two 
of  those  sides. 

454.  Construct  a  triangle,  given  the  median,  the  altitude, 
and  the  angle  bisector,  all  drawn  from  the  same  vertex. 

Hint.     Locate  the  center  of  the  circumscribed  circle. 


62  CONSTRUCTIONS 

455.  Construct  a  triangle,  given  the  feet  of  the  three 
altitudes. 

Hint.  The  altitudes  of  a  triangle  bisect  the  angles  of  the  triangle 
formed  by  joining  their  feet  in  succession. 

456.  If  the  inscribed  and  circumscribed  circles  of  a 
triangle  are  concentric,  the  diameters  are  in  the  ratio 
1:2. 

457.  The  sum  of  the  diameters  of  the  inscribed  and 
circumscribed  circles  of  a  right  triangle  is  equal  to  the 
sum  of  the  legs  of  the  triangle. 

458.  0  is  the  center  of  the  inscribed  circle  of  the  tri- 
angle ABO,  and  0'  the  center  of  the  escribed  circle 
touching  BO;  prove  that  0,  B,  0f,  0,  are  concyclic. 

459.  In  any  triangle,  the  difference  of  two  sides  is  equal 
to  the  difference  of  the  segments  of  the  third  side  formed 
by  the  point  of  tangency  of  the  inscribed  circle. 

460.  0  and  0'  are  the  centers  of  the  inscribed  and 
circumscribed  circles  respectively  of  the  triangle  ABO, 
and  AD  is  perpendicular  to  BO;  prove  that  A  0  bisects 
the  angle  D A 0'. 

461.  The  diagonals  of  a  quadrilateral  ABO  I)  intersect 
at  E;  prove  that  the  centers  of  the  circles  circumscribed 
about  the  triangles  AEB,  BEO,  OED,  DEA,  are  the 
vertices  of  a  parallelogram. 

462.  Construct  a  circle  to  intercept  equal  chords  of  a 
given  length  on  three  given  lines. 

463.  The  radii  of  the  circumscribed  and  escribed  circles 
of  an  equilateral  triangle  are  respectively  double  and  treble 
the  radius  of  the  inscribed  circle. 


CONSTRUCTIONS  63 

464.  Three  circles  whose  centers  are  A,  B,  C,  respec- 
tively, are  tangent  externally,  pair  by  pair,  at  D,  E,  F ; 
prove  that  the  inscribed  circle  of  the  triangle  ABO  is  the 
circumscribed  circle  of  the  triangle  DEF. 

465.  Circumscribe  about  a  given  circle  a  rhombus  whose 
side  is  a. 

What  limits  has  a  ? 

466.  The  area  of  a  circumscribed  square  is  twice  the 
area  of  a  square  inscribed  in  the  same  circle. 

467.  ABOD  is  an  inscribed  square,  and  E  is  any  point 
on  the  circle;  prove  that  EA2+ EB2+ E02+ ED2  is 
twice  the  square  on  the  diameter  of  the  circle. 

468.  Circumscribe  a  square  about  a  given  rectangle,  the 
vertices  of  the  latter  to  lie  on  the  sides  of  the  former. 

469.  Inscribe  a  circle  in  a  quadrant  of  a  given  circle. 

470.  Divide  a  right  angle  into  five  equal  parts. 

Hint.     Divide  the  radius  of  the  circle  in  extreme  and  mean  ratio. 

471.  Two  tangents,  AB,  AC,  are  drawn  to  a  given  circle 
from  A ;  construct  a  circle  tangent  to  AB,  AC,  and  the 
convex  arc  BO. 

472.  From  the  vertex  A  of  an  isosceles  triangle  ABO 
a  line  is  drawn  meeting  the  base  in  D  and  the  circum- 
scribed circle  in  E ;  prove  that  AB  is  tangent  to  the  circle 
circumscribed  about  the  triangle  BBE. 

Hint.  Find  another  angle  equal  to  B  or  to  C.  Draw  the  circum- 
scribed circle  in  question,  and  note  the  measures  of  the  various  angles. 

473.  Twice  the  square  on  a  side  of  an  inscribed  equi- 
lateral triangle  is  equal  to  three  times  the  square  inscribed 
in  the  same  circle. 


64 


CONSTRUCTIONS 


Let  0  be  the  center  of  the  inscribed  circle  of  the  tri- 
angle ABC,  and  0V  02,  0B,  the  centers  of  the  escribed 
circles  tangent  respectively  to  BC,  AC,  AB\    also,  let 

the  three  escribed  circles 
touch  BC  or  BC  pro- 
duced in  Kv  K2,  Kz,  as 
shown  in  the  figure. 
Prove  the  six  following 
theorems :  — 

474.  The  points  A,  0, 
0V  are  collinear;  simi- 
larly, B,  0,  02,  and  C,  0, 
0Z,  are  respectively  col- 
linear. 

475.  The  points  02,  A, 
0S,  are  collinear ;  simi- 
larly,    08,    B,    0V    and 

0V  C,  02,  are  respectively  collinear. 

476.  The  triangles  BOxC,  C02A,  AOsB,  are  similar 
each  to  each. 

Hint.  BOxCO  is  an  inscriptible  quadrilateral,  and  angle  A02C 
equals  angle  OxBC. 

477.  The  triangle  01020z  is  similar  to  the  triangle 
formed  by  joining  the  points  of  tangency  of  the  inscribed 
circle. 

478.  Each  of  the  four  points  0,  0V  02,  0Z,  is  the  ortho- 
center  of  the  triangle  formed  by  joining  the  other  three. 

479.  The  four  circles,  each  of  which  passes  through 
three  of  the  points  0,  0V  02,  08,  are  equal. 

480.  The  orthocenter  and  the  vertices  of  any  triangle 
are  the  centers  of  the  inscribed  and  escribed  circles  of 


CONSTRUCTIONS  65 

the  triangles  formed  by  joining  the  feet  of  the  altitudes 
of  the  given  triangle. 

481.  Given  the  base  and  the  vertical  angle  of  a  tri- 
angle ;  find  the  locus  of  the  center  of  the  inscribed 
circle. 

482.  Given  the  base  and  the  vertical  angle  of  a  tri- 
angle; find  the  locus  of  the  center  of  the  escribed 
circle  which  touches  the  base. 

483.  Given  the  base  and  the  vertical  angle  of  a  trian- 
gle :  prove  that  the  locus  of  the  center  of  the  circum- 
scribed circle  is  a  point. 

484.  The  lines  joining  the  center  of  an  inscribed  circle 
of  a  triangle  to  the  centers  of  the  escribed  circles,  are 
bisected  by  the  circumscribed  circle  of  the  triangle. 

Hint.  Each  line  is  the  hypotenuse  of  a  right  triangle.  Join  the 
vertex  of  the  right  angle  to  the  given  intersection,  and  prove  the  line 
thus  drawn  equal  to  each  of  the  given  segments. 

485.  With  three  given  points  as  centers,  construct  three 
circles  tangent  to  each  other,  pair  by  pair.  How  many 
solutions  are  there  ? 

486.  Given  the  centers  of  the  three  escribed  circles; 
construct  the  triangle. 

487.  Given  the  center  of  the  inscribed  circle  and  of  two 
of  the  escribed  circles ;  construct  the  triangle. 

488.  In  the  triangle  ABC,  the  center  of  the  inscribed 
circle  is  0;  prove  that  the  centers  of  the  circumscribed 
circles  of  the  triangles  A  OB,  BOO,  CO  A,  lie  on  the  cir- 
cumscribed circle  of  ABC. 

Hint.    This  may  be  solved  by  the  aid  of  No.  484. 
conant's  ex.  geom.  — 5 


66  CONSTRUCTIONS 

489.  The  inscribed  circle  of  a  triangle  ABO  is  tangent 
to  BO  at  D  ;  prove  that  the  circles  inseribed  in  the  trian- 
gles ABB,  ACD,  are  tangent  to  each  other. 

490.  In  any  triangle  the  feet  of  the  medians,  the  feet 
of  the  altitudes,  and  the  middle  points  of  the  lines  join- 
ing the  orthocenter  to  the  vertices  are  concyclic. 

Hint.  In  the  triangle  A  BC,  let  X,  Y,  Z,  be  the  middle  points  of 
BC,  AC,  AB,  respectively;  D,  E,  F,  the  feet  of  the  altitudes  drawn 
from  A,  B,  C,  respectively,  0  the  orthocenter,  and  P,  Q,  R,  the  middle 
points  of  A  0,  BO,  CO,  respectively.  Join  Y  and  Z  to  X,  and  each  of 
these  three  points  to  P.  Prove  the  angles  PZX,  PYX,  right  angles, 
and  note  that  the  angle  PDX  is  a  right  angle. 

Note.  This  circle  is  often  called  the  Nine  Points  Circle.  Many 
of  its  properties  are  to  be  derived  from  the  fact  that  it  is  the  circum- 
scribed circle  of  the  triangle  whose  vertices  are  the  feet  of  the  alti- 
tudes of  the  triangle. 

491.  The  center  of  the  nine  points  circle  is  the  middle 
point  of  the  line  joining  the  orthocenter  and  the  center  of 
the  circumscribed  circle. 

Hint.  Letter  the  figure  as  in  the  preceding  problem,  and  let  S  be 
the  center  of  the  circumscribed  circle.  Prove  that  the  perpendicular 
bisectors  of  XD  and  2?  F  pass  through  the  middle  point  of  SO. 

492.  The  radius  of  the  nine  points  circle  is  half  the 
radius  of  the  circumscribed  circle. 

493.  The  centroid  of  a  triangle,  the  center  of  the  cir- 
cumscribed circle,  the  center  of  the  nine  points  circle, 
and  the  orthocenter  of  the  triangle  are  collinear. 

Hint.  With  the  figure  of  No.  491,  let  AX  cut  SO  in  G,  and  prove 
that  AG  is  two  thirds  of  AX. 

494.  The  nine  points  circle  of  the  triangle  ABO,  whose 
orthocenter  is  0,  is  also  the  nine  points  circle  of  each  of 
the  triangles  AOB,  BOO,  00 A, 


CONSTRUCTIONS  67 

495.  All  circles  which  have  the  same  orthocenter  and 
the  same  circumscribed  circle  have  also  the  same  nine 
points  circle. 

Hint.    Prove  by  the  aid  of  Nos.  491  and  492. 

496.  If  four  circles  are  drawn  tangent  to  the  sides  of  a 
quadrilateral,  three  by  three,  their  centers  are  concyclic. 

497.  The  perpendiculars  drawn  from  the  centers  of  the 
three  escribed  circles  of  a  triangle  to  the  sides  they  touch 
are  concurrent. 

498.  Given  an  angle  and  the  radii  of  the  inscribed  and 
circumscribed  circles ;  construct  the  triangle. 

499.  If  the  bisectors  of  the  angles  of  any  polygon  are 
concurrent,  a  circle  may  be  inscribed  in  the  polygon. 

500.  0  is  the  orthocenter  of  the  triangle  ABC,  and  DE 
is  the  diameter  of  the  circumscribed  circle ;  prove  that 

A0*+BC2  =  B02+CA2  =  C02  +  AB2  =  DE2. 


IV.    SIMILAR   FIGURES 

501.  If  three  or  more  lines  divide  any  number  of  par- 
allels into  proportional  parts,  these  lines  are  concurrent. 

State  the  converse  of  this  theorem. 

502.  In  a  circle  a  chord  CD  is  drawn  perpendicular  to 
a  diameter  AB,  and  through  A  any  chord  AE  is  drawn, 
meeting  CD  in  F ;  prove  that  AF  •  AE  is  constant  for  all 
positions  of  AE. 

A  simple  proof  of  the  Pythagorean  proposition  can  be 
obtained  from  this  theorem.  It  is  readily  found  by  mov- 
ing E  along  the  circumference  of  the  circle  until  it  comes 
into  coincidence  with  C. 

503.  Two  lines,  AB,  AC,  are  divided  in  the  points  D,  E, 
respectively,  so  that  AB  •  AD  =  AC-  AE ;  prove  that  the 
points  B,  C,  D,  E,  are  concyclic. 

Hint.  The  proof  depends  on  the  equality  of  the  angles  DBE, 
ECD. 

504.  In  a  triangle  ABC,  AB=S  in.;  if  a  line  parallel 
to  BC  divides  AC  in  the  ratio  of  4:3,  what  are  the  seg- 
ments into  which  it  divides  AB  ? 

505.  The  angles  of  a  triangle  are  30°,  60°,  90°;  find 
the  ratio  of  the  segments  into  which  the  side  opposite  the 
angle  of  60°  is  divided  by  the  bisector  of  that  angle. 

506.  Find  the  legs  of  a  right  triangle  if  their  projections 
on  the  hypotenuse  are  3  ft.  and  6  ft.  respectively. 

507.  In  any  triangle  the  product  of  two  sides  is  equal 

68 


SIMILAR   FIGURES  69 

to  the  product  of  the  diameter  of  the  circumscribed  circle 
and  the  altitude  on  the  third  side. 

Note.  This  theorem  enables  one  to  find  the  radius  of  a  circum- 
scribed circle  when  the  three  sides  of  a  triangle  are  given. 

508.  How  many  miles  at  sea  is  the  light  of  a  lighthouse 
150  ft.  high  visible,  reckoning  the  radius  of  the  earth  as 
3960  miles  ? 

Note.  In  ordinary  problems  of  this  kind  the  distance  in  statute 
miles  to  which  an  object  is  visible,  provided  there  is  no  intervening 
obstacle,  is  approximately  one  and  one  third  times  the  square  root  of 
the  height  of  the  object  in  feet.     A  formula  often  used  by  engineers  is 

K  =  , where  K  is  the  distance  in  statute  miles  and  h  is  the  height 

.7575 

in  feet. 

509.  If  two  similar  triangles  have  their  homologous 
sides  parallel,  the  lines  joining  their  homologous  vertices 
are  concurrent. 

What  happens  when  the  triangles  are  equal? 

510.  In  any  triangle  the  product  of  two  sides  is  equal 
to  the  square  of  the  bisector  of  the  included  angle  plus 
the  product  of  the  segments  into  which  it  divides  the 
third  side. 

Hint.  Produce  the  bisector  to  meet  the  circumscribed  circle,  and 
join  the  point  where  it  meets  the  circle  to  the  extremities  of  the  base. 

Note.  This  theorem  enables  us  to  compute  the  bisectors  of  the 
angles  of  a  triangle  in  terms  of  the  sides  of  the  triangle. 

511.  The  legs  of  a  right  triangle  are  7  ft.  and  8  ft.  re- 
spectively.    Find  their  projections  on  the  hypotenuse. 

512.  Find  the  legs  of  a  right  triangle  if  their  ratio  is 
5  :  4,  and  the  hypotenuse  is  30  ft. 

513.  The  legs  of  a  right  triangle  are  1.564  ft.  and 
2.138  ft.  ;  find  the  segments  of  the  hypotenuse  made  by 
the  bisector  of  the  right  angle. 


70  SIMILAR    FIGURES 

Note.  In  connection  with  these  and  other  problems  involving  the 
right  triangle,  it  is  of  interest  to  know  that  the  ancient  Egyptians 
had,  in  an  empirical  way,  learned  how  to  construct  a  right  angle,  or, 
as  it  is  termed  in  building,  "  a  square  corner."  They  had  found  that 
a  triangle  whose  sides  were  3,  4,  5,  contained  a  square  corner,  and 
made  use  of  that  knowledge  in  squaring  the  corners  of  a  building. 
Three  stakes  were  driven  into  the  ground  in  such  positions  that  a 
rope  passed  around  them  would  form  a  triangle  of  the  required  kind ; 
or  that  an  endless  rope  containing  knots,  so  tied  that  the  segments 
into  which  it  was  divided  were  as  3  : 4  :  5,  would,  when  tightly  stretched 
with  the  points  of  force  applied  at  the  knots,  produce  the  same  effect. 
From  this  fact  builders  were,  in  ancient  Egypt,  sometimes  known  by 
the  curious  name  of  "  rope  twisters,"  or  "  rope  stretchers." 

514.  Find  the  distance  from  the  center  of  a  circle  to  a 
chord  equal  to  the  radius.  What  angle  does  the  sub- 
tended arc  measure  at  the  center  ? 

515.  Two  chords  AB,  CD,  intersect  within  a  circle  at 
E ;  AE  =  10  in.,  BE  =  12  in.,  CD  =  23  in.  Find  CE  and 
DE. 

516.  Show  how  to  find  the  height  of  an  object  situated 
on  the  opposite  bank  of  a  river  from  the  observer. 

Note.  This  problem  involves  finding  also  the  distance  from  the 
observer  to  the  object,  which  is  not  supposed  to  be  known.  In  con- 
nection with  problems  of  this  nature  it  is  interesting  to  note  that  the 
distance  of  an  object  can  be  determined  at  once,  and  with  a  consider- 
able degree  of  accuracy,  by  means  of  an  instrument  known  as  the 
"  range-finder,"  which  is  extensively  used  in  naval  warfare  for  deter- 
mining the  distance  of  a  hostile  ship.  Engineers  also  use  many 
rough  methods  of  approximating  the  distance  of  objects,  of  which 
the  following  are  illustrations :  by  ordinary  eyes  the  windows  of  a 
large  house  can  be  counted  at  a  distance  of  about  13,000  feet,  or 
21  miles ;  men  and  horses  appear  as  points  at  about  half  that  distance ; 
a  horse  can  be  clearly  distinguished  at  about  4000  feet;  the  movemeuts 
of  men  at  2600  feet,  or  about  half  a  mile;  the  head  of  a  man, 
occasionally,  at  2300  feet,  and  very  plainly  at  1300  feet,  or  a  quarter 
of  a  mile.     The  Arabs  of  Algeria  define  a  mile  as  "  the  distance  at 


SIMILAR   FIGURES  71 

which  you  can  no  longer  distinguish  a  man  from  a  woman."  These 
distances  will,  of  course,  vary  with  the  condition  of  the  atmosphere 
and  the  individual  acute ness  of  vision. 

517.  In  any  triangle  the  orthocenter,  the  centroid,  and 
the  intersection  of  the  perpendicular  bisectors  of  the  sides 
are  collinear ;  and  the  distance  between  the  first  two  is 
twice  the  distance  between  the  last  two. 

Hint.  In  the  triangle  ABC,  let  CH,  AK,  be  altitudes,  CM,  AN, 
be  medians,  and  MF,  NF,  perpendicular  bisectors,  locating  the  ortho- 
center  at  D,  the  centroid  at  E,  and  the  intersection  of  the  perpendicu- 
lar bisectors  at  F.  The  triangles  A  DC,  MNF,  are  similar,  and  the 
sides  of  the  former  are  double  the  sides  of  the  latter.  Prove  that  the 
triangles  A  DE,  NEF,  are  similar. 

518.  The  sides  of  a  triangle  are  8,  11,  13 ;  find  the  seg- 
ments of  the  sides  made  by  the  respective  angle  bisectors. 

519.  At  the  extremities  of  a  line  AB  perpendiculars 
AB,  BE,  are  drawn,  and  in  AB,  or  AB  produced,  a  point 
O  is  taken  such  that  the  angle  DC  A  is  equal  to  the  angle 
EAB.  If  AB  =  25  in.,  AD=  13  in.,  and  BE=1  in.,  find 
OA  and  OB. 

520.  In  any  inscribed  quadrilateral  the  product  of  the 
diagonals  is  equal  to  the  sum  of  the  products  of  the  oppo- 
site sides. 

Hint.  Let  A  BCD  be  the  quadrilateral.  In  the  diagonal  AC 
take  a  point  E  such  that  the  angle  EDC  shall  equal  the  angle  ADB. 
Two  pairs  of  similar  triangles  can  now  be  found. 

521.  In  any  triangle  the  sum  of  the  squares  of  any  two 
sides  is  equal  to  twice  the  square  of  half  the  third  side 
plus  twice  the  square  of  the  median  drawn  to  the  third 
side. 

Note.  By  means  of  this  theorem  the  medians  can  be  computed 
when  the  sides  are  known. 


72  SIMILAR   FIGURES 

522.  If  E  and  F  are  the  middle  points  of  the  sides  AB, 
CD,  respectively,  of  the  parallelogram  ABCD,  and  AF  and 
OF  be  drawn  intersecting  the  diagonal  BD  in  H  and  L 
respectively,  and  BF  and  BE  be  drawn  intersecting  the 
diagonal  AC  in  K  and  Q-  respectively,  then  is  GHKL  a 
parallelogram  equal  in  area  to  one  ninth  of  ABCD, 

523.  The  sum  of  the  squares  of  the  sides  of  a  parallelo- 
gram is  equal  to  the  sum  of  the  squares  of  the  diagonals. 

524.  The  sum  of  the  squares  of  the  diagonals  of  a  trape- 
zoid is  equal  to  the  sum  of  the  squares  of  the  legs  plus 
twice  the  product  of  the  bases. 

Hint.     Join  the  middle  points  of  the  diagonals  and  use  No.  521. 

525.  Every  straight  line  cutting  the  sides  of  a  triangle, 
or  the  sides  produced,  determines  upon  the  sides  six  seg- 
ments such  that  the  product  of  three  non-consecutive  seg- 
ments is  equal  to  the  product  of  the  other  three. 


Hint.  The  cutting  line  DEF  may  cut  two  sides  of  the  triangle 
and  the  third  side  produced,  or  the  three  sides  produced.  The  proof 
is  the  same  in  either  case.  The  segments,  and  the  equation  between 
them,  are  as  follows :  — 

ADBE.CF=AF.BD>CE. 

The  proof  is  obtained  from  the  proportions  existing  between  the  sides 
of  the  similar  triangles. 


SIMILAR   FIGURES  73 

This  theorem  is  due  to  Menelaus  of  Alexandria.  It  was  discovered 
about  80  B.C. 

526.  Lines  drawn  through  the  vertices  of  a  triangle 
determine,  if  concurrent,  six  segments  on  the  sides  of  the 
triangle  such  that  the  product  of  three  non-consecutive 
segments  is  equal  to  the  product  of  the  other  three. 

Hint.  The  lines  may  meet  within  or  without  the  triangle.  Use 
No.  525  on  each  of  the  two  parts  into  which  one  of  the  line's  divides 
the  triangle.     This  theorem  was  invented  by  Ceva  of  Milan  in  1678. 

527.  The  hypotenuse  of  a  right  triangle  is  13,  and  the 
sum  of  the  legs  is  17  ;  find  the  legs. 

528.  Find  the  legs  of  a  right  triangle  if  their  sum  is  49 
and  the  sum  of  their  squares  is  1881. 

529.  The  radii  of  two  circles  are  7  in.  and  4  in. 
respectively,  and  the  distance  between  the  centers  of  the 
circles  is  12  in. ;  find  the  length  of  the  common  exterior 
tangent. 

Can  you  think  of  a  problem,  similar  to  the  above,  in 
which  the  length  of  the  tangent  mentioned  shall  equal 
the  diameter  of  one  of  the  circles? 

530.  In  an  isosceles  triangle  the  base  is  13  in.  and 
each  leg  is  11  in. ;  find  the  radius  of  the  circumscribed 
circle. 

Hint.     See  No.  507. 

531.  Through  a  point  A  without  a  circle,  a  tangent 
AB  and  a  secant  AOD  are  drawn;  AC  =8  in.,  and 
CD  =  5  in.;   find  AB. 

532.  Through  a  point  A,  8  in.  from  the  center  of  a 
circle  whose  radius  is  4  in.,  a  secant  ABC  is  drawn  such 
that  BC=2  in.;  find  AB. 


74  SIMILAR  FIGURES 

533.  The  sides  of  a  triangle  are  432  in.,  586  in.,  and 
728  in.  respectively  ;  find  the  length  of  the  shortest 
median.     (See  No.  521.) 

To  which  side  of  a  triangle  is  the  shortest  median 
drawn  ?  the  longest  median  ? 

534.  The  medians  drawn  from  the  extremities  of  the 
hypotenuse  of  the  right  triangle  AB Care  BE,  CF ;  prove 
that  4  BE2  +  4  OF2  =  5  BO2. 

535.  Any  point  E  within  the  rectangle  ABCJD  is  joined 
to  each  of  the  four  vertices  ;  prove  that  AE2  -f  CE2  = 
BE2  +  DE2. 

Examine  this  theorem  for  the  case  when  E  lies  without 
the  rectangle  ;  on  the  perimeter. 

536.  Every  quadrilateral  is  divided  by  its  diagonals  into 
four  triangles  whose  areas  are  proportional  to  each  other. 

Hint.  From  two  opposite  vertices,  drop  perpendiculars  to  the 
same  diagonal. 

537.  If  one  of  the  parallel  sides  of  a  trapezoid  is  double 
the  other,  the  diagonals  intersect  at  a  point  of  trisection. 

Hint.    Make  one  of  the  diagonals  the  median  of  a  triangle. 

538.  In  a  side  A 0  of  a  triangle  ABC  any  point  D  is 
taken,  and  the  lines  AB,  DC,  AB,  BC,  are  bisected  at  the 
points  E,  F,  Q-,  H,  respectively ;  prove  that  EG  =  FH. 

539.  The  area  of  a  ring  bounded  by  two  concentric 
circles  is  equal  to  the  area  of  a  circle  whose  diameter  is  a 
chord  of  the  outer  circle  tangent  to  the  inner  circle. 

How  can  this  area  be  found  by  using  the  circle  which 
lies  midway  between  the  two  given  circles? 

540.  The  radius  of  a  circle  is  a  mean  proportional 
between  the  segments  of  any  tangent  made  by  its  point  of 
contact  and  any  pair  of  parallel  tangents. 


SIMILAR   FIGURES.  75 

541.  AB  is  a  diameter  of  a  circle,  and  through  A  any 
straight  line  is  drawn  to  cut  the  circumference  at  0  and 
the  tangent  at  B  in  D ;  prove  that  A  0  is  a  third  propor- 
tional to  AD  and  AB. 

Write  out  the  proportion  which  results  when  the  limit- 
ing case  is  reached,  and  state  whether  or  not  it  is  still 
true. 

542.  Within  a  regular  hexagon  whose  side  is  10  in.  a 
second  regular  hexagon  is  inscribed  by  joining  the  middle 
points  of  the  sides  taken  in  order.  What  is  the  area  of 
the  inscribed  hexagon? 

Will  the  area  of  the  inscribed  hexagon  remain  the  same 
if  any  other  points  in  the  outer  hexagon  are  chosen,  the 
inscribed  hexagon  still  remaining  regular? 

543.  At  the  extremities  of  a  diameter  of  a  circle  tan- 
gents are  drawn  meeting  a  third  tangent  in  A,  B  ;  if  C  is 
the  point  of  contact  of  the  third  tangent,  prove  that 
A  0  •  OB  is  constant  for  all  positions  of  O. 

Hint.     Prove  by  means  of  No.  540. 

544.  A  swimmer  whose  eye  is  on  the  level  of  the  water 
can  just  see  the  top  of  a  stake  8  in.  high  at  a  distance  of 
a  mile.     What  is  the  diameter  of  the  earth  ? 

545.  If  similar  polygons  are  constructed  on  the  three 
sides  of  a  right  triangle  as  homologous  sides  of  the  poly- 
gons, the  polygon  drawn  on  the  hypotenuse  is  equal  in 
area  to  the  sum  of  the  polygons  drawn  on  the  other  two 
sides. 

What  celebrated  name  in  mathematics  and  philosophy 
is  suggested  by  this  theorem  ? 

546.  ABC  is  an  isosceles  triangle  whose  vertex  is  A; 
on  the  base,  or  the  base  produced,  any  point  D  is  taken  ; 


76  SIMILAR   FIGURES 

prove  that  the  circumscribed  circles  of  the  triangles  ABD, 
ACD,  are  equal. 

Hint.     Prove  by  means  of  the  equal  angles  of  the  given  triangle. 

547.  Find  the  locus  of  a  point  whose  distance  from  two 
intersecting  lines  is  as  m  :  n. 

Hint.  The  locus  consists  of  two  straight  lines  through  the  inter- 
section of  the  two  giveu  lines. 

548.  ABC  and  DEF  are  two  similar  triangles  whose 
areas  are  respectively  245  sq.  in.  and  5  sq.  in.  If  AB  = 
21  in.,  find  DE,  the  homologous  side  of  DEF. 

What  answer  would  you  obtain  if  the  above  figures  were 
quadrilaterals  instead  of  triangles,  the  given  dimensions 
remaining  the  same  ? 

549.  Through  any  point  A  without  a  circle,  secants  are 
drawn  to  the  circle.  Find  the  locus  of  the  point  which 
divides  the  entire  secants  in  the  ratio  of  m  :  n. 

Hint.  Divide  the  line  joining  A  with  the  center  of  the  circle  in 
the  ratio  of  m  :  n. 

550.  A  quarter-mile  running  track  is  to  be  laid  out 
with  straight  parallel  sides  and  semicircular  ends.  The 
track  is  to  be  10  ft.  wide,  and  the  distance  between  the 
outer  parallel  edges  is  to  be  220  ft.  What  must  be  the 
extreme  length  of  the  field  in  which  the  track  is  to  be 
laid  out,  so  that  a  runner  may  cover  a  quarter  of  a  mile 
by  keeping  in  the  middle  of  the  track?  By  keeping  close 
to  the  inner  edge  of  the  track  ? 

551.  The  radii  of  two  intersecting  circles  are  10  in.  and 
17  in.  respectively,  and  the  distance  between  their  centers 
is  21  in.     Find  the  length  of  their  common  chord. 

552.  The   parallel   sides   of  a   circumscribed  isosceles 


SIMILAR  FIGURES  77 

trapezoid  are  18  in.  and  6  in.  respectively.     What  is  the 
radius  of  the  circle  ? 

Hint.     Extend  the  non-parallel  sides  until  they  meet. 

553.  Find  the  locus  of  a  point  such  that  the  sum  of  the 
squares  of  its  distances  from  two  given  points  is  constant. 

The  locus  is  a  circle  whose  center  is  at  the  middle  point 
of  the  line  connecting  the  two  points.     (See  No.  521.) 

554.  Find  the  locus  of  a  point  such  that  the  difference 
of  the  squares  of  its  distances  from  two  given  points  is 
constant. 

The  locus  consists  of  two  parallel  straight  lines,  which 
are  perpendicular  to  the  line  joining  the  two  given  points. 

555.  The  fly  wheel  of  an  engine  is  connected  by  a  belt 
with  a  smaller  wheel  driving  the  machinery  of  a  mill. 
The  radius  of  the  fly  wheel  is  7  ft.  and  of  the  driving 
wheel  21  in.  How  many  revolutions  does  the  small 
wheel  make  to  one  of  the  large  wheel  ?  The  distance 
between  the  centers  is  10  ft.  6  in.  What  is  the  length  of 
the  belt  connecting  the  two  wheels  ? 

556.  Through  any  point  A  a  line  is  drawn  cutting  a 
given  circle  in  B,  O;  if  J.  moves  so  that  the  product  of 
the  segments  AB,  AC,  is  constant,  find  the  locus  of  the 
point  A.     (Two  cases.) 

What  is  the  locus  if  AB  •  AC=  0  ?     If  AB  •  AC=  r2? 

Hint.  When  A  is  within  the  circle,  draw  through  A  a  chord  per- 
pendicular to  the  line  joining  A  to  the  center  of  the  circle. 

557.  Divide  a  line  32  in.  long  into  three  parts,  which 
shall  be  to  each  other  as  \ :  f :  2^. 

558.  Divide  a  line  so  that  the  product  of  its  segments 
shall  equal  a  given  quantity. 


78  SIMILAR   FIGURES 

559.  What  is  the  area  of  a  trefoil  formed  on  an  equi- 
lateral triangle  whose  side  is  6  in.  ? 

Note.  If  from  each  of  the  vertices  of  a  regular  polygon  as  a 
center,  with  a  radius  equal  to  half  the  side  of  the  polygon,  a  circle 
be  described  outwardly,  an  ornamental  figure  extensively  used  in 
architecture  will  be  obtained.  If  the  polygon  be  a  triangle,  the 
figure  is  called  a  trefoil;  if  a  square,  a  quatrefoil;  if  a  pentagon,  a 
cinquefoil.  A  figure  of  this  kind  formed  about  a  polygon  of  many 
sides  is  often  used ;  it  is  called  a  rose  window. 

560.  A  rose  window  of  six  lobes  is  to  be  placed  in  a 
circular  opening  35  ft.  in  diameter.  How  many  square 
feet  of  glass  will  it  contain,  no  deduction  being  made  for 
sash  or  leading  ? 

561.  If  a  circle  be  tangent  internally  to  another  circle 
of  twice  its  radius,  the  path  of  a  given  point  on  the  cir- 
cumference of  the  smaller  as  it  rolls  about  within  the 
larger,  always  remaining  tangent  to  it,  is  a  diameter  of 
the  larger  circle. 

What  about  the  velocity  of  the  moving  point,  the  rate 
of  the  moving  circle  being  constant  ? 

562.  If  a  given  square  be  subdivided  into  n2  equal 
squares,  n  being  any  given  number,  and  a  circle  be 
inscribed  in  each  of  these  equal  squares,  the  sum  of 
these  circles  is  equal  in  area  to  the  circle  inscribed  in 
the  original  square. 

563.  Divide  a  line  so  that  the  product  of  the  whole 
line  and  one  of  the  segments  shall  equal  a  given  quantity. 

Hint.  This  problem  readily  reduces  to  the  problem  of  finding  a 
third  proportional  to  two  given  quantities. 

564.  Construct  two  lines,  given  their  difference  and 
their  ratio. 

Note.  Care  should  be  taken  to  work  with  lines,  not  merely  with 
algebraic  values. 


SIMILAR   FIGURES  79 

565.  Find  the  locus  of  a  point  whose  distances  from 
two  given  points  are  in  the  ratio  m :  n. 

Hint.  Draw  through  the  two  points  a  line  of  indefinite  length, 
and  determine  on  this  line  two  points  of  the  locus.  The  entire  locus 
is  a  circle  passing  through  the  two  points.  This  problem  has  many- 
important  applications  in  certain  parts  of  physics.  Of  these  the 
following  is  an  example :  — 

566.  Find  the  locus  of  a  point  in  a  plane  equally  illu- 
minated by  two  lights  in  the  plane  ;  given,  that  the  inten- 
sities of  the  lights  at  a  unit's  distance  are  as  m  :  w,  and  that 
the  intensity  of  any  light  varies  inversely  as  the  square 
of  the  distance. 

567.  Find  the  locus  of  a  point  from  which  a  given  line 
is  seen  so  as  to  subtend  a  given  angle. 

568.  Find  the  locus  of  the  vertex  of  a  triangle,  having 
given  the  base  and  the  ratio  of  the  other  two  sides. 

569.  Through  any  given  point  draw  a  line  which  shall 
cut  the  sides  of  a  given  angle  so  that  the  segments  be- 
tween the  vertex  and  the  points  of  intersection  shall  be  in 
a  given  ratio.     (Three  cases.) 

570.  Find  in  one  side  of  a  triangle  a  point  whose  dis- 
tances from  the  other  two  sides  shall  be  in  a  given  ratio. 
(See  No.  547.) 

571.  Find  within  a  triangle  a  point  whose  distances 
from  the  three  sides  of  the  triangle  shall  be  as  m  :  n  :  p. 

Can  such  a  point  be  found  without  the  triangle  ? 

572.  Construct  a  circle  having  a  given  radius,  touching 
a  given  circle,  and  having  the  distances  from  its  center  to 
two  given  lines  in  a  given  ratio. 

To  the  above  problem  apply  the  method  of  loci,  and  dis- 
cuss the  number  of  solutions. 


80  SIMILAR   FIGURES 

573.  Find  a  point  such  that  it  shall  be  equidistant  from 
two  given  lines,  and  its  distances  from  two  given  points 
shall  be  in  a  given  ratio. 

574.  The  ground  plan  of  a  house  drawn  on  a  scale  of 
J  in.  to  1  ft.  is  represented  by  a  rectangle  8J  in.  by  1  ft.  ; 
what  are  the  dimensions  of  the  foundation  of  the  house  ? 

575.  A  field  containing  9  acres  is  represented  by  a  tri- 
angular plan  whose  sides  are  12  in.,  17  in.,  and  25  in.  On 
what  scale  is  the  plan  drawn? 

576.  Find  a  point  whose  distances  from  two  given  points 
shall  be  in  a  given  ratio,  and  whose  distances  from  two 
other  given  points  shall  also  be  in  a  given  ratio. 

577.  Find  the  position  of  a  point  which  is  equally  illu- 
minated by  three  lights  of  unequal  intensities  situated  in 
the  same  plane. 

578.  A  triangle  ABC  is  divided  into  three  parts  which 
are  of  equal  area  by  lines  drawn  parallel  to  AB ;  if  BO  is 
100  inches  long,  find  the  length  of  each  of  the  segments 
into  which  it  is  divided  by  the  lines  parallel  to  AB. 

579.  In  a  given  triangle  insci  be  a  rectangle  similar  to 
a  given  rectangle. 

o  Z^^f 

5r^— Uf-" """""  i 


H    FB 

Hint.  Let  ABC  be  the  given  triangle  and  P  the  given  rectangle. 
On  the  altitude  CH  construct  a  rectangle  CL  similar  to  the  given 
rectangle.  The  line  A K  will  determine  a  point  E  which  will  be  one 
of  the  vertices  of  the  required  rectangle.     Why  ? 

580.  In  the  triangle  ABC  draw  a  line  parallel  to  AB 
cutting  AC,  BC,  in  2),  E,  so  that  AB  :  DE=  m  :  n. 


SIMILAR  FIGURES  81 

581.  Perform  the  above  construction  so  that  AB  :  BU= 
BE  :  BO. 

582.  In  a  given  circle  inscribe  a  triangle  similar  to  a 
given  triangle. 

583.  In  a  given  square  whose  side  is  16  in.  a  square  is 
inscribed  by  joining  the  middle  points  of  the  sides  taken 
in  order ;  in  this  square  another  square  is  inscribed  in  a 
similar  manner,  and  so  on.  Find  the  area  of  the  first  in- 
scribed square ;  of  the  eighth  inscribed  square. 

Hint.  This  problem  can  readily  be  converted  into  a  problem 
which  involves  the  finding  of  the  last  term  of  a  geometrical  progres- 
sion. 

584.  Find  the  locus  of  a  point  such  that  the  tangents 
drawn  from  it  to  two  given  circles  shall  be  equal. 

The  locus  is  a  line  perpendicular  to  the  line  of  centers, 
and  is  called  the  radical  axis  of  the  two  circles ;  and  the 
required  point  is  the  intersection  of  this  locus  with 
the  given  line.  The  point  where  the  radical  axis  cuts 
the  line  of  centers  is  a  point  which  divides  the  line  of 
centers  into  two  segments,  the  difference  of  whose 
squares  is  equal  to  the  difference  of  the  squares  of  the 
radii  of  the  circles. 

If  the  two  circles  are  tangent  to  each  other,  the  radical 
axis  is  the  common  interior  tangent;  and  if  the  circles 
intersect,  it  is  the  common  chord. 

The  radical  axis  is  also  the  locus  of  the  centers  of  cir- 
cles intersecting  the  two  given  circles  at  right  angles. 

585.  Find  in  a  given  line  a  point  such  that  the  tangents 
drawn  from  it  to  two  given  circles  shall  be  equal. 

586.  In  a  given  triangle  a  similar  triangle  is  inscribed 
by  joining  the  middle  points  of  the  sides  ;  in  this  inscribed 

conant's  ex.  geom.  —  6 


82  SIMILAR   FIGURES 

triangle  another  triangle  is  inscribed  in  the  same  manner, 
and  so  on.  What  fraction  of  the  given  triangle  is  the 
area  of  the  sixth  inscribed  triangle? 

587.  In  a  circle  whose  radius  is  32  in.  an  equilateral 
triangle  is  inscribed,  in  this  triangle  a  circle,  in  this  circle 
another  equilateral  triangle,  and  so  on.  Find  the  area  of 
the  third  inscribed  circle.  Which  circle  has  an  area  of 
3.14159? 

588.  Construct  a  triangle,  given  two  of  the  angles,  and 
the  sum  of  the  altitude  and  the  median  drawn  from  the 
same  vertex. 

In  many  constructions  like  the  above,  it  is  easy  to  con- 
struct a  figure  similar  to  the  required  figure,  and  then  to 
construct  the  required  figure  by  means  of  the  familiar 
theorem  that  in  similar  figures  homologous  lines  are  pro- 
portional. In  this  example,  draw  a  triangle  similar  to  the 
required  triangle,  draw  the  altitude  and  median  homolo- 
gous to  the  altitude  and  median  whose  sum  is  given,  and 
extend  the  altitude  beyond  the  base  by  a  length  equal  to 
the  median  of  the  triangle  now  drawn.  The  remainder  of 
the  construction  can  then  be  obtained  by  means  of  the 
general  theorem  just  suggested. 

589.  Construct  a  triangle,  given  two  angles,  and  the 
difference  between  the  radii  of  the  inscribed  and  circum- 
scribed circles. 

Are  the  radii  of  the  inscribed  and  circumscribed  cir- 
cles of  two  similar  triangles  homologous  lines  of  those 
triangles  ? 

590.  Construct  a  triangle,  given  two  angles,  and  the 
sum  of  the  bisector  of  the  third  angle  and  the  side  oppo- 
site that  angle. 


SIMILAR   FIGURES  83 

591.  Construct  a  triangle,  given  one  angle,  the  ratio 
between  the  adjacent  sides,  and  the  sum  of  the  third  side 
and  the  altitude  let  fall  upon  that  side. 

592.  Construct  a  triangle,  given  the  ratio  of  one  side  to 
each  of  the  other  sides,  and  the  radius  of  the  inscribed 
circle. 

593.  Construct  a  rectangle,  given  the  ratio  between  one 
of  the  longer  sides  and  a  diagonal,  and  the  sum  of  a  diag- 
onal and  one  of  the  shorter  sides. 

594.  Construct  a  triangle,  having  given  the  three  alti- 
tudes. 

Hint.  Any  two  altitudes  of  a  triangle  are  inversely  proportional 
to  the  sides  upon  which  they  are  let  fall. 

595.  Construct  a  parallelogram,  having  given  an  angle, 
the  sum  of  the  diagonals,  and  the  ratio  between  two  adja- 
cent sides. 

596.  In  a  given  circle,  draw  a  chord  which  shall  be  tri- 
sected by  two  given  radii. 

597.  Construct  a  chord  quadrilateral,  having  given  two 
opposite  sides,  one  angle,  and  the  angle  between  the  diag- 
onals. 

598.  Construct  a  quadrilateral,  given  one  diagonal,  and 
the  four  angles  which  the  other  diagonal  makes  with  the 
sides. 

599.  Construct  a  quadrilateral,  given  the  sum  of  the 
diagonals,  and  the  four  angles  which  one  of  the  diagonals 
makes  with  the  sides. 

600.  Construct  a  triangle,  given  two  angles,  and  the  sum 
of  the  included  side  and  the  angle  bisector  drawn  upon 
that  side. 


V.  AREAS  OF  FIGURES.  EQUAL  AREAS 

601.  Two  triangles  or  two  parallelograms  having  two 
adjacent  sides  respectively  equal  and  the  included  angles 
supplementary  have  equal  areas. 

Compare  the  area  of  a  triangle  and  that  of  a  parallelo- 
gram under  the  conditions  of  the  problem. 

602.  The  areas  of  two  mutually  equiangular  parallelo- 
grams are  to  each  other  as  the  products  of  two  adjacent 
sides. 

603.  Find  the  side  of  a  square  whose  area  is  equal  to 
that  of  an  equilateral  triangle  whose  side  is  6  in. 

Which  will  have  the  greater  perimeter  ? 

604.  The  area  of  a  rectangular  field  is  |  of  an  acre,  and 
its  length  is  double  its  breadth ;  if  a  horse  is  picketed  at 
the  middle  point  of  one  of  the  longer  sides,  find  the  length 
of  the  picket  rope  which  will  enable  him  to  graze  over 
half  the  field. 

605.  A  circle  whose  radius  is  8  in.  has  half  its  area  re- 
moved by  cutting  a  ring  away  around  the  outside;  find 
the  width  of  the  ring. 

606.  The  perimeter  of  a  square  is  72  ft.,  and  the  perim- 
eter of  a  rectangle  whose  length  is  twice  its  breadth  is 
also  72  ft.     Find  the  difference  between  their  areas. 

If  two  figures,  square  and  rectangle,  have  equal  areas, 
which  has  the  lesser  perimeter?  What  other  figures  of 
equal  area  can  be  mentioned  which  have  a  still  smaller 
perimeter  ?     Which  of  them  has  the  least  ? 

84 


AREAS  85 

607.  Find  the  area  of  a  rectangle  if  its  diagonal  is  50  m. 
and  its  sides  are  in  the  ratio  3:5. 

608.  The  dimensions  of  a  rectangle  are  64  ft.  and  58  ft. 
respectively.  If  the  length  is  diminished  by  10  ft.,  how 
much  must  the  breadth  be  increased  in  order  that  the*area 
may  remain  the  same  ? 

609.  The  perimeter  of  a  rhombus  is  20  in.  and  its  shorter 
diagonal  is  6  in. ;  find  the  side  of  a  square  of  the  same  area. 

Is  the  side  of  the  required  square  greater  or  less  than 
the  diameter  of  an  equivalent  circle  ? 

610.  Each  of  the  acute  angles  of  a  rhombus  is  60°  ;  find 
the  ratio  of  its  area  to  that  of  a  square  whose  perimeter  is 
equal  to  the  perimeter  of  the  rhombus. 

As  the  acute  angles  of  the  rhombus  decrease,  its  perim- 
eter remaining  the  same,  how  does  its  area  change  ? 

611.  If  two  parallelograms  have  equal  areas,  and  the 
altitude  of  one  is  four  times  the  altitude  of  the  other,  what 
is  the  ratio  of  their  bases  ? 

What  would  the  ratio  be  if  the  figures  were  triangles 
instead  of  parallelograms  ? 

612.  The  bases  of  two  triangles  are  equal  and  their 
altitudes  are  6  in.  and  8  in.  respectively;  what  is  the 
ratio  of  their  areas? 

What  would  the  ratio  be  if  the  above  figures  were  par- 
allelograms instead  of  triangles  ? 

613.  The  altitudes  of  two  triangles  are  equal,  and  their 
bases  are  2  in.  and  3  in.  respectively ;  what  is  the  base  of  a 
triangle  whose  area  is  equal  to  the  sum  of  their  areas,  and 
whose  altitude  is  half  as  great  as  their  common  altitude  ? 

If  in  the  above  problem  the  last-named  figure  were  a 
parallelogram,  what  would  be  its  base  ? 


86  AREAS 

614.  One  leg  of  a  right  triangle  is  12  m.  and  the  alti- 
tude on  its  hypotenuse  is  10  m. ;  find  the  area  of  the  tri- 
angle. 

In  what  two  ways  can  the  hypotenuse  be  found  ? 

615.  The  area  of  a  triangle  is  68  sq.  yd. ;  find  its  base 
and  altitude  if  they  are  as  12  :  7. 

616.  What  is  the  area  of  a  triangle  if  the  perimeter  is 
21  in.  and  the  radius  of  the  inscribed  circle  is  1.6  in.? 

617.  Find  the  area  of  the  greatest  circle  that  can  be 
cut  out  of  a  triangle  whose  sides  are  6  in.,  8  in.,  10  in. 
long  respectively. 

Is  the  perimeter  of  a  triangle  any  index  of  the  area  of 
the  inscribed  circle  ?  of  the  circumscribed  circle  ?     Why  ? 

618.  What  is  the  area  of  a  trapezoid  if  its  altitude  is 
8  ft.  and  its  median  is  18  ft.  ? 

Can  a  circle  be  inscribed  in  a  trapezoid  ?  circumscribed 
about  a  trapezoid  ? 

619.  A  trapezoid  contains  450  sq.  ft.  and  its  altitude  is 
20  ft. ;  find  the  bases  of  the  trapezoid  (i)  if  one  is  4  ft. 
longer  than  the  other ;   (ii)  if  they  are  in  the  ratio  3 : 4. 

620.  The  value  of  a  field  in  the  shape  of  a  trapezoid  is 
16000.  The  bases  are  120  yd.  and  220  yd.  respectively, 
and  the  altitude  is  206  yd. ;  what  is  the  value  of  the  land 
per  acre  ? 

621.  The  base  of  a  triangle  is  150  ft.  and  its  altitude  is 
180  ft.  ;  find  the  base  of  a  parallelogram  of  equal  area 
whose  altitude  is  52  ft. 

622.  Upon  the  diagonal  of  a  rectangle  20  ft.  long  and 
18  ft.  wide  a  triangle  is  constructed  whose  area  is  equal 
to  the  area  of  the  rectangle ;  calling  this  line  the  base  of 
the  triangle,  what  must  be  its  altitude  ? 


AREAS  87 

623.  Find  the  perimeter  of  a  rhombus  whose  shorter 
diagonal  is  6  in.,  if  its  area  is  equal  to  that  of  a  triangle 
whose  base  is  12  in.  and  whose  altitude  is  14  in. 

624.  Prove  that  the  area  of  a  regular  dodecagon  is 
three  times  the  square  of  the  radius  of  the  circumscribed 
circle. 

625.  Find  the  side  of  a  regular  hexagon  whose  area  is 
equal  to  the  area  of  an  equilateral  triangle  whose  side  is 
12  ft. 

If  the  areas  of  two  rhombuses  are  equal,  are  their 
perimeters  necessarily  equal  ? 

626.  Find  a  point  in  one  side  of  a  triangle  such  that 
a  line  drawn  through  it  parallel  to  one  of  the  other  sides 
shall  bisect  the  triangle. 

Having  found  such  a  point,  does  it  make  any  difference 
which  of  the  remaining  sides  is  chosen  as  the  one  to  which 
the  required  line  shall  be  drawn  parallel  ? 

627.  Find  the  side  of  an  equilateral  triangle  whose  area 
is  equal  to  the  area  of  a  square  having  a  diagonal  6  in.  in 
length. 

628.  Find  the  side  of  a  square  equal  in  area  to  a  quad- 
rilateral whose  sides  are  3  ft.,  4  ft.,  5  ft.,  6  ft.,  respectively, 
and  one  of  whose  diagonals  is  7  ft. 

In  the  arrangement  of  the  sides  of  this  quadrilateral, 
place  the  sides  opposite  each  other  which  are  3  ft.  and  5  ft. 
respectively.  What  other  arrangement  is  possible  ?  Is 
the  area  the  same  in  both  cases  ? 

629.  How  must  a  line  parallel  to  the  base  of  a  triangle 
be  drawn  so  as  to  cut  off,  in  that  part  of  the  triangle  next 
the  vertex,  (i)  two  fifths  of  the  area  of  the  triangle  ? 
(ii)  three  eighths  of  the  area  ? 


88  AREAS 

630.  Find  the  side  of  a  rhombus  composed  of  two  equi- 
lateral triangles,  and  equal  in  area  to  a  rhombus  whose 
diagonals  are,  respectively,  1  ft.  3  in.  and  1  ft.  8  in. 

631.  Through  the  vertex  of  a  triangle  whose  area  is 
100  sq.  ft.,  a  line  is  drawn  dividing  it  into  two  parts,  one 
containing  12  sq.  ft.  more  than  the  other;  what  must 
be  the  length  of  each  of  the  segments  into  which  the  base 
is  divided  by  this  line,  if  the  entire  length  of  the  base  is 
14  ft.  ? 

632.  In  a  triangle  ABC  the  side  AB  =  7  in.,  and  the 
side  A 0=  9  in.  On  AB  a  point  D  is  taken  4  in.  from  A, 
and  BE  is  drawn,  cutting  the  side  AC  in  E  so  that  the 
triangle  ABE  is  two  sevenths  of  AB  0.  Find  the  length 
of  the  segment  AE. 

If  two  sides  of  a  triangle  are  known,  how  much  do  we 
know  about  its  area  ? 

633.  How  must  lines  be  drawn  through  the  middle 
point  of  one  of  the  sides  of  a  triangle  so  as  to  divide  its 
area  into  three  parts  which  are  to  each  other  as  2 :  3  :  7  ? 

634.  In  the  base  AB  of  the  triangle  ABC,  the  point  D 
is  50  ft.  from  A  and  175  ft.  from  B.  The  triangle  is  to 
be  divided  into  three  parts,  which  are  as  2 :  3  :  5,  by  lines 
BE,  BE,  drawn  from  D  to  cut  AC,  BC,  respectively. 
Find  AE,  BF,  in  terms  of  AC,  BC,  respectively. 

Hint.  Apply  the  theorem  which  proves  that  triangles  having 
equal  altitudes  are  to  each  other  as  their  bases. 

635.  The  base  AB  of  a  triangle  is  50  ft.  long,  and  D, 
the  foot  of  the  altitude  from  the  vertex  C,  is  40  ft.  from 
A  ;  find  a  point  E  in  AB  such  that  the  perpendicular 
erected  at  E  will  divide  the  triangle  into  (i)  equivalent 
parts  ;   (ii)  parts  in  the  ratio  of  3  :  4. 


AREAS  89 

Hint.  Obtain  three  proportions  involving  three  unknown  quan- 
tities, by  means  of  the  theorem  that  triangles  are  to  each  other  as 
the  products  of  their  bases  and  altitudes. 

636.  The  area  of  a  parallelogram  is  80  sq.  ft.;  how 
must  a  line  from  one  vertex  divide  one  of  the  opposite 
sides  in  order  that  the  triangle  cut  off  may  contain  30 
sq.  ft.  ? 

637.  What  part  of  a  parallelogram  is  contained  between 
one  half  of  one  side  and  one  third  of  one  of  the  adjacent 
sides  ? 

Does  it  make  any  difference  which  of  the  adjacent 
sides  is  chosen? 

638.  In  a  trapezoid  ABCD,  the  bases  are  AB=6  ft., 
CD  =  4  ft. ;  find  a  point  E  in  the  diagonal  DB,  such  that 
a  line  through  _Z7,  parallel  to  AD,  will  divide  the  trapezoid 
into  two  parts,  which  shall  be  to  each  other  as  3 :  4. 

639.  The  base  and  altitude  of  a  triangle  are  28  ft.  and 
22  ft.  respectively ;  find  the  area  of  a  triangle  formed 
by  drawing  a  line  parallel  to  the  base  and  6  ft.  from  the 
vertex. 

640.  The  sides  of  two  equilateral  triangles  are  3  ft. 
and  4  ft.  respectively;  find  the  side  of  an  equilateral 
triangle  equivalent  to  their  sum. 

641.  Find  the  area  of  a  square  if  the  sum  of  a  diagonal 
and  a  side  is  14  in. 

642.  The  altitude  of  a  triangle  is  8  in. ;  what  is  the 
homologous  altitude  of  a  similar  triangle  ten  times  as 
large  ?     Of  a  similar  triangle  n  times  as  large  ? 

643.  One  side  ^  of  a  triangle  is  a ;  find  the  homologous 
side  of  a  similar  triangle  one  third  as  large.  One  nth  as 
large. 


90  AREAS 

644.  One  side  of  a  polygon  is  3  in.  ;  find  the  homolo- 
gous side  of  a  similar  polygon  having  to  the  given  polygon 
a  ratio  of  3 :  5. 

645.  If  the  side  of  one  equilateral  triangle  is  equal  to 
the  altitude  of  another,  find  the  ratio  of  their  areas. 

646.  Transform  a  given  triangle  into  an  equivalent 
rhombus  with  a  given  length  a  for  one  of  its  diagonals. 

Does  it  make  any  difference  which  diagonal  is  chosen 
equal  to  a? 

647.  Trisect  a  parallelogram  by  lines  drawn  from  one 
of  the  vertices. 

648.  The  diagonals  of  two  squares  are  3  ft.  and  4  ft. 
respectively ;  find  the  diagonal  of  a  square  equal  in  area 
to  their  sum. 

649.  The  sides  of  two  equilateral  triangles  are  2  ft.  and 
5  ft.  respectively ;  find  the  side  of  an  equilateral  triangle 
whose  area  is  equal  to  their  difference. 

650.  The  sides  of  a  triangle  are  10  in.,  16  in.,  and  20  in. 
respectively ;  find  the  area  of  each  of  the  two  parts  into 
which  the  triangle  is  divided  by  the  bisector  of  the  angle 
between  the  first  two  sides. 

651.  The  base  of  a  triangle  is  6  in.  and  its  altitude  is 
8  in. ;  find  the  change  in  area  if  these  dimensions  are : 
(i)  increased  by  3  in.  and  2  in.  respectively ;  (ii)  decreased 
by  3  in.  and  2  in.  respectively ;  (iii)  one  increased  by  3  in. 
and  the  other  diminished  by  the  same  amount. 

652.  Transform  a  given  triangle  into  an  equivalent 
isosceles  triangle  with  a  given  length  a  for  one  of  its  legs. 

653.  Construct  a  triangle  five  times  as  large  as  a  given 
triangle  ;  one  fifth  as  large. 


AREAS  91 

How  many  ways  are  there  of  performing  this  con- 
struction ? 

654.  Construct  a  triangle  equivalent  to  (i)  the  sum  of 
two  triangles  with  equal  altitudes ;  (ii)  the  difference  of 
two  triangles  with  equal  bases. 

655.  Find  the  change  in  area  of  a  triangle  of  base  16  in. 
and  altitude  18  in.  :  (i)  if  the  base  is  increased  by  5  in. 
and  the  altitude  is  diminished  by  4  in. ;  (ii)  if  the  base  is 
decreased  by  5  in.  and  the  altitude  is  increased  by  4  in.  ; 
(iii)  if  the  base  is  increased  by  5  in.  and  the  altitude  is 
diminished  by  5  in. 

656.  Transform  a  given  triangle  into  an  isosceles  tri- 
angle of  equal  area,  whose  altitude  shall  equal  a. 

657.  Construct  a  square  equivalent  to  three  sevenths  of 
a  given  square. 

658.  Transform  a  triangle  into  an  equivalent  triangle 
with  its  altitude  equal  to  a,  and  the  angle  from  which  this 
altitude  is  drawn  equal  to  a  given  angle  6. 

659.  Transform  a  given  triangle  into  an  equivalent  tri- 
angle with  base  and  altitude  equal  to  each  other,  and  a 
given  length  a  for  the  median  drawn  to  the  base. 

660.  Transform  a  given  triangle  into  an  equivalent  tri- 
angle similar  to  another  given  triangle. 

Hint.  The  base  of  the  required  triangle  is  a  mean  proportional 
between  the  base  of  the  first  triangle  and  a  triangle  which  is  similar 
to  the  second  given  triangle,  and  which  has  an  altitude  equal  to  the 
altitude  of  the  first  triangle. 

661.  Divide  a  circle  by  one  or  more  concentric  circum- 
ferences into 

(i)  Two  equivalent  parts. 
(ii)  Four  equivalent  parts. 


92  AREAS 

(iii)  Two  parts  of  which  the  inner  shall  be  three  times 
the  outer. 

662.  Transform  a  square  into  an  equivalent  rectangle 
having  a  given  perimeter. 

Hint.     Form  two  equations  containing  two  unknown  quantities. 

663.  Transform  a  square  into  an  equivalent  rectangle 
having  a  given  diagonal. 

664.  Transform  a  rectangle  into  an  equivalent  rectangle 
such  that  the  difference  between  two  of  its  adjacent  sides 
shall  equal  a  given  quantity  a. 

665.  Transform  a  parallelogram  into  an  equivalent 
rhombus  having  for  one  of  its  diagonals  a  side  of  the 
parallelogram. 

666.  Inscribe  in  a  given  circle  a  rectangle  whose  area 
is  equal  to  the  area  of  a  given  square. 

Hint.     Form  two  equations  containing  two  unknown  quantities. 

667.  Transform  a  given  polygon  into  an  equivalent 
polygon  similar  to  another  given  polygon. 

Hint.     First  transform  the  two  given  polygons  into  squares. 

668.  Construct  a  polygon  equivalent  to  the  sum  of  two 
given  polygons  and  similar  to  another  given  polygon. 

669.  Divide  a  trapezoid  into  five  parts  whose  areas  shall 
be  equal,  each  to  each. 

670.  Divide  a  trapezium  into  seven  parts  whose  areas 
shall  be  equal,  each  to  each. 

Hint.  Draw  one  of  the  diagonals  and  divide  it  into  seven  equal 
parts. 

671.  Divide  a  triangle  into  three  parts  which  shall  be 
to  each  other  as  2  :  3 :  4. 

Can  this  division  be  effected  by  more  than  one  method? 


AREAS  93 

672.  Divide  a  triangle  into  two  equivalent  parts  by  a 
line  drawn  through  a  point  in  one  side.  Divide  the  tri- 
angle into  five  equivalent  parts  by  lines  drawn  through  a 
point  in  the  longest  side. 

Hint.  To  bisect  the  triangle  ABC,  take  any  point  D  in  the  side 
AB  as  the  point  through  which  the  bisecting  line  is  to  be  drawn. 
From  C  draw  a  line  to  M,  the  middle  point  of  AB,  and  connect  CD. 
Then  the  required  line  DE  can  be  found  by  drawing  ME  parallel 
to  CD. 

673.  Find  a  point  within  a  triangle  such  that  the  lines 
drawn  from  this  point  to  the  three  vertices  shall  divide 
the  triangle  into  three  parts  whose  areas  shall  be  equal, 
each  to  each. 

Hint.     Draw  the  medians  of  the  triangle. 

674.  Divide  a  triangle  into  five  equivalent  parts  by 
lines  drawn  from  a  given  point  within  the  triangle. 

Hint.  Connect  the  given  point  with  one  of  the  vertices  of  the 
triangle,  and  begin  by  cutting  off  on  either  side  of  this  line  a  triangle 
or  a  quadrilateral  as  the  case  may  be,  whose  area  is  equal  to  one 
fifth  the  area  of  the  original  triangle.  This  is  a  general  method,  and 
is  applicable  where  no  special  method  of  solution  can  be  found. 

675.  Divide  a  triangle  into  two  equivalent  parts  by  a 
line  perpendicular  to  one  of  its  sides. 

676.  Divide  a  triangle  into  four  equivalent  parts  by 
lines  parallel  to  one  of  the  angle  bisectors. 

677.  Divide  a  triangle  into  three  equivalent  parts  by 
lines  parallel  to  one  of  the  medians. 

678.  Divide  a  parallelogram  into  five  equivalent  parts 
by  lines  drawn  from  one  of  its  vertices. 

679.  Divide  a  trapezoid  into  three  equivalent  parts  by 
lines  drawn  from  one  of  its  vertices. 

Hint.     First  divide  the  trapezoid  into  two  equivalent  parts. 


94  AREAS 

680.  Divide  a  parallelogram  into  two  parts  which  shall 
be  to  each  other  as  2  :  3  by  a  line  drawn  from  a  given  point 
in  one  side. 

681.  Bisect  a  parallelogram  by  a  line  drawn  through 
any  given  point. 

Hint.  What  point  is  the  center  of  a  parallelogram  ?  Has  a  triangle 
a  center?    A  trapezoid? 

682.  Bisect  a  parallelogram  by  a  line  parallel  to  any 
given  line. 

683.  Bisect  a  trapezoid  by  a  line  parallel  to  the  bases. 
(See  hint  to  No.  662.) 

684.  Bisect  a  trapezoid  by  a  line  drawn  from  any  given 
point  in  one  of  the  bases  of  the  trapezoid. 

685.  Bisect  a  trapezoid  by  a  line  parallel  to  any  given  line. 

686.  Divide  a  hexagon  into  two  parts  which  shall  be  in 
the  ratio  of  2  :  3  by  a  line  drawn  through  a  given  point  in 
one  of  the  sides.     (See  hint  to  No.  674.) 

687.  Inscribe  in  a  triangle  a  rectangle  which  shall  have 
a  given  area.     (See  hint  to  No.  662.) 

688.  Inscribe  in  a  given  parallelogram  a  rhombus  which 
shall  have  a  given  area. 

Hint.  Using  the  values  of  the  diagonals  of  the  rhombus,  form 
two  equations  as  in  No.  662.  Note  the  relation  of  the  center  of  the 
rhombus  to  the  center  of  the  parallelogram. 

689.  ABO  is  any  triangle,  and  D  is  any  point  in  A B  ; 
draw  through  B  a  line  BE  to  meet  BC  produced  in  h\ 
so  that  the  triangle  BBE  shall  be  equivalent  to  the 
triangle  ABO. 

690.  On  a  base  of  given  length  construct  a  triangle 
equivalent  to  any  given  triangle,  and  having  its  vertex  on 
a  given  line. 


AREAS  95 

691.  Construct  a  parallelogram  having  the  same  area 
and  the  same  perimeter  as  a  given  triangle. 

Is  there  more  than  one  construction  ?  Is  the  construction 
always  possible  ?    Will  the  required  figure  ever  be  a  square  ? 

692.  Bisect  a  trapezium  by  a  line  drawn  through  one  of 
its  vertices. 

693.  On  the  base  AD  of  any  quadrilateral  ABCD  con- 
struct a  triangle  whose  area  shall  be  equal  to  the  area  of 
the  quadrilateral,  and  having  the  base  and  the  angle  A 
coincide  with  the  base  and  the  angle  A  of  the  quadrilateral. 

694.  Cut  off  from  a  given  quadrilateral  a  third,  a  fourth, 
a  fifth,  or  any  given  fraction  of  its  area  by  a  straight  line 
drawn  through  one  of  its  vertices. 

695.  Construct  a  circle  equivalent  to  a  given  triangle. 

696.  On  a  given  base  construct  a  triangle  whose  area  is 
equal  to  the  area  of  a  given  circle. 

697.  Construct  a  circle  whose  area  is  equal  to  the  area 
of  a  given  quadrilateral. 

698.  Construct  a  quadrilateral  whose  area  is  equal  to 
the  area  of  a  given  circle. 

Can  more  than  one  quadrilateral  be  constructed  that 
will  fulfill  the  given  conditions  ? 

699.  A  circular  field  is  300  ft.  in  diameter,  and  a  horse 
is  picketed  by  a  rope  20  ft.  long,  the  picket  pin  being 
driven  exactly  on  the  boundary  of  the  field  ;  over  how 
many  square  feet  can  the  horse  graze 

(i)  Inside  the  field  ? 
(ii)  Outside  the  field  ? 

700.  Find  the  center  of  gravity  of  any  convex  quadri- 
lateral. 


VI.     MISCELLANEOUS   THEOREMS   AND 
PROBLEMS 

701.  Divide  a  line  internally  and  externally  so  that  the 
segments  may  be  in  a  given  ratio. 

Def.  A  straight  line  divided  internally  and  externally 
into  segments  having  the  same  ratio  is  said  to  be  divided 
harmonically. 

702.  If  AB  be  divided  harmonically  at  P,  Q,  where  Q  is 
the  external  point,  then  is  AB  the  harmonic  mean  between 
AQfm&AP. 

703.  If  AB  be  divided  harmonically  at  P,  Q,  where  Q  is 
the  external  point,  and  0  is  the  middle  point  of  AB,  then 
is  CP-CQ=  CA\ 

704.  Two  villages  are  on  opposite  sides  of  a  river. 
Show  by  construction  how  a  bridge  should  be  located  in 
order  that  the  distance  from  a  selected  point  in  each  village 
to  the  nearer  end  of  the  bridge  shall  be  the  same  in  each 
case. 

For  convenience  suppose  the  banks  of  the  river  to  be 
straight  and  parallel,  and  the  ground  level.  Examine  the 
problem  for  different  locations  of  the  villages. 

DEFINITIONS 

If  AB  be  divided  harmonically  at  P,  Q,  then  are  P,  Q% 
the  harmonic  conjugates  of  A,  B.  In  this  case  the  points 
A,  2?,  are  also  harmonic  conjugates  of  P,  Q. 


MISCELLANEOUS   EXERCISES 


97 


Any  series  of  points  in  a  straight  line  is  called  a  range. 

Four  points  so  placed  that  two  of  them  are  harmonic 
conjugates  of  the  other  two  are  said  to  form  a  harmonic 
range. 

If  any  number  of  straight  lines  intersect  at  a  common 
point,  these  lines  are  said  to  form  a  pencil. 

Each  of  the  lines  which  form  a  pencil  is  called  a  ray. 

The  point  of  divergence  of  the  rays  of  a  pencil  is  called 
the  vertex  of  the  pencil. 

A  pencil  of  four  rays  which  pass  through  the  four  points 
of  a  harmonic  range  respectively  is  called  a  harmonic  pencil. 

A  system  of  four  straight  lines,  none  of  which  are  con- 
current, intersect  with  each  other  to  form  a  figure  which 
is  called  a  complete  quadrilateral.  Such  a  quadrilateral  has 
six  vertices  and  three  diagonals. 

In  the  following  figure  we  have  a  system  of  four  straight 
lines,  none  of  which 
are  concurrent. 

These  lines  form 
by  their  inter- 
section the  com- 
plete quadrilateral 
ABODEF.  The 
six  vertices  are  in- 
dicated by  the  six 
letters,  and  the  di- 
agonals, which  are 
not  drawn  in  the 
figure,  are  the  lines 
AB,ED,CF.  Any 
ordinary  quadrilateral  can  be  transformed  into  a  complete 
quadrilateral  by  extending  the  opposite  pairs  of  sides  until 
they  meet.  If  either  pair  of  opposite  sides  consists  of 
conant's  ex.  geom.  —  7 


98  MISCELLANEOUS   EXERCISES 

parallel  lines,  these  lines  are  said  to  meet  and  to  form  a 
vertex  at  infinity,  or  at  an  infinite  distance  from  the  other 
vertices  of  the  quadrilateral. 

705.  If  a  line  be  drawn  parallel  to  any  ray  of  a  harmonic 
pencil,  the  other  three  rays  intercept  equal  parts  on  it. 

706.  Any  transversal  is  cut  harmonically  by  the  rays  of 
a  harmonic  pencil. 

707.  The  bisector  of  the  angle  formed  by  a  pair  of  con- 
jugate rays  is  perpendicular  to  its  own  conjugate  ray. 

708.  If  the  homologous  sides  of  any  two  similar,  unequal 
figures  are  parallel,  each  to  each,  the  lines  joining  corre- 
sponding vertices  are  concurrent ;  and  the  distances  from 
any  pair  of  homologous  vertices  to  the  point  of  concur- 
rence are  in  the  same  ratio  as  any  pair  of  homologous  sides. 

Does  this  theorem  undergo  any  modification  when  the 
two  figures  are  equal  ? 

Def.  The  point  of  concurrence  of  the  lines  connecting 
the  homologous  vertices  of  two  similar  figures  so  placed 
that  their  homologous  sides  are  parallel  is  cajled  the  center 
of  similitude. 

709.  A  secant  intersects  two  given  circles,  and  meets 
the  line  of  centers  at  its  point  of  intersection  with  the 
common  tangent.  Prove  that  the  radii  drawn  to  the 
points  of  intersection  of  the  secant  and  the  circle  are 
parallel,  pair  by  pair. 

710.  In  the  above  construction,  let  A,  A\  be  the  points 
of  tangency,  B  the  point  of  intersection  of  the  tangent 
and  the  line  of  centers,  and  (7,  D,  C ',  D1 ',  the  points  of 
intersection  of  the  secant  and  the  circles,  taken  in  order 
from  B ;  to  prove  that  BA  -  BA'  =  BC  •  BD'  =  BD  •  BC. 


MISCELLANEOUS   EXERCISES  99 

Def.  The  points  which  divide  externally  and  inter- 
nally the  line  of  centers  of  two  circles  in  the  ratio  of  their 
radii  are  called  the  direct  and  the  inverse  centers  of  simili- 
tude respectively. 

The  definition  just  given  is  readily  seen  to  cover  the 
special  case,  that  which  applies  to  circles  only,  of  the 
more  general  definition  of  center  of  similitude  given 
above. 

711.  If  a  variable  circle  touches  two  fixed  circles,  the 
line  which  passes  through  the  two  points  of  contact 
passes  also  through  a  center  of  similitude.  How  many 
cases? 

712.  The  two  radii  of  a  circle  drawn  to  its  points  of 
intersection  with  any  line  passing  through  a  center  of 
similitude  are  parallel,  pair  by  pair,  to  the  two  radii 
of  the  other  circle  drawn  to  its  intersections  with  the 
same  line. 

713.  The  common  exterior  tangents  of  two  circles 
pass  through  the  direct  center  of  similitude,  and  the 
common  interior  tangents  pass  through  the  inverse  cen- 
ter of  similitude. 

What  method  of  drawing  the  common  tangents  to  two 
circles  does  this  theorem  suggest?  Is  there  any  other 
simple  method  of  performing  this  construction  ? 

*  714.  In  a  given  sector  of  a  circle,  inscribe  a  square  so 
that  two  of  the  vertices  of  the  square  shall  be  on  the  arc, 
and  the  other  two  on  the  radii  of  the  sector. 

POLES   AND  POLARS 

Def.  If  on  any  line  drawn  from  the  center  of  a  circle 
two  points  are  taken  such  that  the  product  of  their  dis- 


100  MISCELLANEOUS   EXERCISES 

tances  from  the  center  is  equal  to  the  square  of  the  radius, 
each  of  these  points  is  called  the  inverse  of  the  other. 

Def.  If  through  either  of  two  inverse  points  a  line  is 
drawn  perpendicular  to  the  line  joining  the  inverse  points 
with  the  center,  the  line  thus  drawn  is  called  the  polar  of 
the  other  point  with  respect  to  the  circle ;  and  the  latter 
point  is  called  the  pole  of  the  polar  thus  drawn. 

A  simple  construction  will  show  that  the  polar  of  an 
external  point  cuts  the  circle ;  the  polar  of  an  internal 
point  lies  wholly  without  the  circle  ;  and  the  polar  of  a 
point  on  the  circle  is  tangent  to  the  circle  at  that  point. 

715.  The  polar  with  respect  to  a  circle  of  an  external 
point  is  the  chord  of  contact  of  the  tangents  drawn  from 
that  point  to  the  circle. 

716.  If  J.,  B,  are  any  two  points  such  that  the  polar 
of  A  with  respect  to  a  given  circle  passes  through  B, 
the  polar  of  B  with  respect  to  the  same  circle  passes 
through  A. 

717.  The  locus  of  the  intersection  of  tangents  drawn 
through  the  extremities  of  all  chords  passing  through  a 
given  point  is  the  polar  of  that  point  with  respect  to  the 
circle. 

718.  Any  straight  line  drawn  through  a  point  is  cut 
harmonically  by  the  point,  its  polar  with  respect  to  an^ 
circle,  and  the  circumference  of  that  circle. 

The  preceding  theorems  show  that 

(i)  The  polar  of  an  external  point  with  respect  to  a 
circle  is  the  chord  of  contact  of  tangents  from  that  point. 

(ii)  The  polar  of  an  internal  point  is  the  locus  of  the 
intersection  of  tangents  drawn  at  the  extremities  of  all 
chords  passing  through  it. 


MISCELLANEOUS   EXERCISES  10J. 

(iii)  The  polar  of  a  point  on  a  circle  is  the  tangent  at 
that  point. 

719.  The  point  of  intersection  of  the  polars  of  two 
given  points  with  respect  to  any  circle  is  the  pole  of  the 
line  passing  through  the  given  points. 

720.  Find  the  locus  of  the  poles  with  respect  to  a  given 
circle  of  all  straight  lines  passing  through  a  given  point. 

721.  Find  the  locus  of  poles  with  respect  to  a  given 
circle  of  tangents  drawn  to  a  concentric  circle. 

722.  Any  two  points  subtend  at  the  center  of  a  circle 
an  angle  equal  to  one  of  the  angles  formed  by  the  polars 
of  the  given  points. 

723.  If  two  circles  intersect  orthogonally,  the  center  of 
each  is  the  pole  of  their  common  chord  with  respect  to  the 
other  circle. 

724.  Any  two  points  subtend  at  the  center  of  a  circle 
an  angle  equal  to  one  of  the  angles  formed  by  the  polars 
of  the  two  given  points. 

725.  Given  any  point  0  on  a  fixed  straight  line  AB ;  find 
the  locus  of  the  point  inverse  to  0  with  respect  to  the  circle. 

726.  The  polars  with  respect  to  a  given  circle  of  the 
four  points  of  a  harmonic  range  form  a  harmonic  pencil. 

Def.  The  radical  axis  of  two  circles  is  the  locus  of  a 
point  so  situated  that  tangents  drawn  from  any  point  of 
the  locus  to  the  two  circles  are  equal. 

727.  Prove  that  the  radical  axis  of  two  circles  is  a  line 
perpendicular  to  the  line  of  centers,  such  that  the  differ- 
ence of  the  squares  of  the  segments  of  the  line  of  centers 
is  equal  to  the  difference  of  the  squares  of  the  radii  of  the 
given  circles. 


102  MISCELLANEOUS  EXERCISES 

728.  Draw  the  radical  axis  of  two  given  circles.  If  the 
circles  intersect,  what  does  the  radical  axis  become  ? 

729.  The  radical  axes  of  three  circles  taken  pair  by 
pair  are  concurrent. 

Dep.  The  point  of  intersection  of  the  radical  axes  of 
three  or  more  circles  is  called  the  radical  center. 

730.  The  radical  axis  of  any  pair  of  circles  bisects  any 
one  of  their  four  common  tangents. 

731.  If  tangents  are  drawn  to  two  circles  from  any 
point  in  their  radical  axis,  a  circle  with  this  point  as  a 
center  and  either  tangent  as  radius  cuts  both  the  given 
circles  orthogonally. 

732.  If  tangents  to  three  circles  are  drawn  from  their 
radical  center,  a  circle  whose  center  is  the  radical  center 
of  the  three  circles  and  whose  radius  is  any  tangent  thus 
drawn  cuts  all  the  circles  orthogonally. 

733.  If  three  circles  are  tangent  to  each  other,  pair  by 
pair,  their  common  tangents  at  their  points  of  contact 
are  concurrent. 

734.  If  circles  are  drawn  on  the  three  sides  of  a  triangle 
as  diameters,  their  radical  center  is  the  orthocenter  of  the 
triangle. 

735.  All  circles  passing  through  a  fixed  point  and  cut- 
ting a  given  circle  orthogonally  pass  through  a  second 
fixed  point. 

736.  Find  the  locus  of  the  centers  of  all  circles  which 
pass  through  a  given  point  and  cut  a  given  circle  orthog- 
onally. 

737.  Find  the  locus  of  the  centers  of  all  circles  which 
cut  two  given  circles  orthogonally. 


MISCELLANEOUS   EXERCISES  103 

738.  Given  a  line  and  a  point  without  the  line.  Draw 
a  line  passing  through  the  point  and  perpendicular  to  the 
given  line 

(i)    When  the  point  is  accessible  and  the  line  is  not. 
(ii)  When  the  line  is  accessible  and  the  point  is  not. 

739.  If  two  triangles  have  one  angle  of  one  equal  to 
one  angle  of  the  other,  and  a  second  angle  of  one  supple- 
mentary to  a  second  angle  of  the  other,  the  sides  about 
the  third  angles  are  proportional. 

740.  AB  bisects  the  angle  A  of  the  triangle  ABO,  and 
meets  the  base  at  D ;  prove  that  if  circles  are  circum- 
scribed about  the  triangles  ABB,  AOB,  their  diameters 
are  to  each  other  as  the  segments  of  the  base. 

741.  AB  and  AE  bisect  internally  and  externally  the 
vertical  angle  A  of  a  triangle,  meeting  the  base  at  B,  E, 
respectively  ;  if  F  is  the  middle  point  of  BO,  prove  that 
FB  is  a  mean  proportional  between  FD  and  FE. 

742.  In   the   triangle   ABO,  AO=2BO;    if  OB,   OE, 

bisect  the  angle  O  internally  and  externally,  meeting  AB 
in  Z>,  E,  respectively,  prove  that  the  areas  of  the  triangles 
OBB,  AOB,  ABO,  OBE  are  as  1  :  2  :  3  :  4. 

743.  How  can  the  area  of  a  triangular  field  be  found 
when  one  side  of  the  field  cannot  be  measured  ?  Give 
more  than  one  method. 

744.  If  through  the  middle  point,  of  the  base  of  a 
triangle  any  line  be  drawn  intersecting  one  side  of  the 
triangle,  the  other  side  produced,  and  the  line  drawn 
from  the  vertex  parallel  to  the  base,  it  will  be  divided 
harmonically. 

745.  If  from  either  base  angle  of  a  triangle  a  line  be 


104  MISCELLANEOUS   EXERCISES 

drawn  intersecting  the  median  from  the  vertex,  the  oppo- 
site side,  and  the  line  drawn  from  the  vertex  parallel  to 
the  base,  it  will  be  divided  harmonically. 

746.  Any  transversal  is  divided  harmonically  by  two 
sides  of  a  triangle  and  the  internal  and  external  bisectors 
of  the  angle  included  by  these  sides. 

THE  PROBLEM  OF   APOLLONIUS 

The  ten  following  problems  are  closely  related,  and 
form  a  series  of  which  the  last  is  called  the  problem  of 
Apollonius.  It  was  first  solved  by  the  famous  Greek 
geometer  of  that  name  about  200  B.C. 

747.  Construct  a  circle  which  shall  pass  through  three 
given  points. 

748.  Construct  a  circle  which  shall  pass  through  two 
given  points  and  be  tangent  to  a  given  line. 


Hint.  Since  AB2  =  AP  x  AP',  the  point  B  can  be  determined 
as  soon  as  A  has  been  found.  As  the  figure  shows,  there  are  two 
solutions. 

749.  Construct  a  circle  which  shall  pass  through  two 
given  points  and  be  tangent  to  a  given  circle. 

Hint.  Let  A ,  B,  be  the  given  points  and  the  circle  whose  center  is  0 
be  the  given  circle.     Draw  any  circle,  with  its  center  at  0',  which  shall 


MISCELLANEOUS   EXERCISES 


105 


pass  through  A ,  B,  and  intersect  the  given  circle  in  two  points,  as  C,  D. 
Draw  a  line  connecting  CD  and  extend  it  to  intersect  the  line  joining 


A,  B,  in  some  point  E.  Draw  tangents  from  E  to  the  circle  0,  touching 
at  FF'.  The  required  circle  is  the  circle  passing  through  the  points 
A,  By  F,  or  through  the  points  A,  B,  F'.  There  are  two  solutions. 
Can  there  ever  be  more  than  two?    Can  there  ever  be  less  than  two? 

750.    Construct  a  circle  which  shall  pass  through  a  given 
point  and  be  tangent  to  two  given  lines. 


d- 

Hint.  Let  A  be  the  given  point  and  BC,  BD,  be  the  given  lines. 
Construct  any  circle  tangent  to  both  lines  and  lying  within  that  angle 
formed  by  the  two  lines,  in  which  A  lies.     Let  S  be  the  center  of  the 


106 


MISCELLANEOUS   EXERCISES 


circle  thus  drawn.  Draw  BA,  SF,  SO.  Then  the  center  of  the 
required  circle  is  determined  by  a  line  AO  or  AO',  drawn  from  A 
parallel  to  FS  or  to  GS,  intersecting  the  bisector  of  the  angle  CBD 
in  0  or  in  0'.  There  are  two  solutions.  Can  there  ever  be  more  or 
less  than  two? 

751.    Construct  a  circle  which  shall  pass  through  a  given 
point  and  be  tangent  to  a  given  line  and  to  a  given  circle. 


Hint.  Ket  A  be  the  given  point  and  BC  the  given  line,  and  the 
circle  whose  center  is  S  the  given  circle.  Suppose  the  problem  solved, 
and  let  the  circle  whose  center  is  0  be  the  required  circle.  Draw 
SD1.BC  and  SO  connecting  the  centers  of  the  given  and  of  the 
required  circles.  Draw  FH  from  F  through  the  point  of  tangency 
of  the  two  circles,  and  prolong  it  to  L.  Draw  the  line  A  GF,  and 
connect  HE  and  OL.  It  can  now  be  shown  that  OL  and  FD  are 
parallel,  that  FA  x  FG  =  FH  x  FL  =  FE  x  FD,  and  that  the  aux- 
iliary circle  whose  center  is  K,  drawn  through  the  points  D,  E,  A,  will 
also  pass  through  G.  This  determines  a  second  point  on  the  required 
circle,  and  reduces  the  problem  to  No.  748. 

The  maximum  number  of  solutions  is  four;  two  are  obtained  l>v 
connecting  A  with  F  and  two  by  connecting  A  with  E.  The  center 
of  one  of  the  latter  is  O '.  Are  there  ever  less  than  four  solutions? 
Is  the  problem  ever  impossible? 


MISCELLANEOUS    EXERCISES 


107 


752.  Construct  a  circle  which  shall  pass  through  a  given 
point  and  be  tangent  to  two  given  circles. 

Hint.  Let  the  circles  whose  centers  are  S  and  Sf  be  the  given 
circles,  and  A  the  given  point.  Suppose  the  problem  solved,  and  let 
the  circle  whose  center  is  O  be  the  required  circle.  Draw  a  line 
through  S  and  S',  and  also  a  line  through  G,  H,  the  points  where  the 
required  circle  touches  the  given  circles.  Let  these  lines  intersect 
at  L.  This  point  is  the  center  of  similitude  of  the  given  circles.  Pass 
a  circle  through  the  three  points  D,  C,  A,  and  it  will  intersect  the 


required  circle  at  a  second  point  M.  A  line  drawn  through  A,  M, 
will  pass  through  L.  Hence  the  problem  can  be  reduced  to  No.  749 
by  drawing  the  auxiliary  circle  through  the  points D,  C,  A,  thus  deter- 
mining a  second  point  on  the  required  circle,  namely,  the  point  of 
intersection  of  the  auxiliary  circle  and  the  line  connecting  A  with 
the  center  of  similitude  of  the  given  circles. 

Is  the  problem  ever  impossible  ?  How  many  solutions  are  there  ? 
Is  the  number  of  solutions  always  the  same? 

753.  Construct  a  circle  which  shall  be  tangent  to  three 
given  lines. 

Is  the  number  of  solutions  to  this  problem  always  the 
same? 


108  MISCELLANEOUS   EXERCISES 

754.  Construct  a  circle  which  shall  be  tangent  to  two 
given  lines  and  to  a  given  circle. 

Hint.  Let  A  G,  AH,  be  the  given  lines,  and  the  circle  whose  center 
is  S  the  given  circle.  Draw  lines  parallel  to  AG,  AH,  as  indicated 
in  the  figure,  at  a  distance  from  them  equal  to  the  radius  of  the  given 
circle.  Then  by  No.  750  construct  a  circle  which  shall  pass  through 
S  and  be  tangent  to  the  two  parallels  thus  drawn.  The  center  of 
this  circle  0  is  also  the  center  of  the  required  circle. 


This  problem  should  be  fully  discussed  for  different  cases,  arising 
from  different  relative  positions  of  the  circle  and  the  lines,  and  the  num- 
ber of  solutions  noted.     The  maximum  number  of  solutions  is  eight. 

755.  Construct  a  circle  which  shall  be  tangent  to  a 
given  line  and  to  two  given  circles. 

Hint.  Let  AB  be  the  given  line,  and  the  circles  whose  centers  are 
S  and  S',  the  given  circles.  Draw  a  line  parallel  to  .  1 B  al  a  distance 
from  it  equal  to  the  radius  of  the  smaller  circle,  letting  S  be  the 
center  of  that  circle,  and  with  S'  as  a  center  and  a  radius  equal  to 
the  difference  of  the  radii  of  the  two  given  circles,  construct  a  circle 
concentric  with  the  greater  of  the  two  given  circles.  By  No.  751 
construct  a  circle  which  shall  pass  through  5  and  be  tangent  to  CD, 


MISCELLANEOUS   EXERCISES 


109 


and  to  the  auxiliary  circle  whose  center  is  S'.     The  center  of   the 
circle  thus  found,  0,  is  also  the  center  of  the  required  circle.     This 


gives  a  single  construction  under  a  special  set  of  conditions.  The 
problem  should  be  discussed  for  different  relative  positions  of  the 
line  and  circles,  and  also  for  the  use  of  the  sum  instead  of  the  differ- 
ence of  the  radii  of  the  given  circles  as  the  radius  of  the  auxiliary 
circle.     The  maximum  number  of  solutions  is  eight. 

756.  Construct  a  cir- 
cle which  shall  be  tan- 
gent to  three  given 
circles.  (The  problem 
of  Apollonius.) 

Hint.  Let  the  circles 
whose  centers  are  S,  S',  S", 
be  the  given  circles,  and  let 
their  radii  be  r,  r',  r",  respec- 
tively, with  r"  >r'>r.  Con- 
struct a  circle  whose  center 
is  Sr  and  whose  radius  is 
r  —  r',  and  a  circle  whose 
center  is  S"'  and  whose  ra- 
dius is  r"  —  r.  Then  by 
No.  752  construct  a  circle  which  shall  pass  through  S  and  be  tangent 


110  MISCELLANEOUS   EXERCISES 

to  these  two  auxiliary  circles.     The  center  of  the  circle  0  is  also  the 
center  of  the  required  circle. 

By  using  all  possible  combinations  of  the  sums,  as  well  as  the 
differences  of  the  radii  of  the  given  circles,  the  maximum  number  of 
solutions  is  found  to  be  eight. 

SOLUTION   OF   GEOMETRIC    PROBLEMS  BY  THE  ALGE- 
BRAIC METHOD 

757.  Construct  a?,  when  x  =  a  +  b  —  c. 

7 

758.  Construct  x  =  — 

c 

Hint.    Make  x  a  fourth  proportional 

759.  Construct  x  =  a  V2. 

760.  Construct  x  =  a^s/5. 

761.  Construct  x  =  — , 

Va2  +  b2 


762.  Construct  x  =  Va2  +  b2  —  ab. 

Hint.  Construct  a  triangle  two  of  whose  sides  are  a  and  b,  and 
whose  included  angle  is  60° ;  x  is  the  altitude  of  this  triangle. 

763.  Construct  the  roots  of  the  equation  x2  —  ax  -+-  b2  =  0. 

Hint.  Since  the  sum  of  the  roots  of  any  quadratic  equation, 
x2  +  px  +  q  —  0,  is  equal  to  —  p,  and  the  product  of  the  roots  is  equal 
to  q,  the  problem  is  at  once  reduced  to  the  following  problem  :  To 
construct  two  lines,  given  their  sum  a  and  their  product  b2. 

764.  Construct  the  roots  of  the  equation  x2  +  ax  +  b2  =  0. 

765.  Construct  the  roots  of  the  equation  x2  —  ax  —  b2  =  0. 

Hint.  In  this  case  the  roots  must  have  contrary  signs,  since 
their  product,  &2,  is  negative.  Hence  it  can  readily  be  seen  that  this 
problem  reduces  to  the  following:  To  construct  two  lines,  given  their 
difference  a  and  their  product  b2.  To  find  these  lines  remember  that 
if  a  tangent  and  a  secant  are  drawn  to  a  circle  from  any  external 


MISCELLANEOUS   EXERCISES  111 

point,  the  tangent  is  a  mean  proportional  between  the  whole  secant 
and  its  external  segment. 

766.  Construct  the  roots  of  the  equation  x2  +  ax  —  b2  =  0. 

Note.  Every  complete  quadratic  equation  can  be  reduced  to  one 
of  the  four  forms  given  in  Nos.  762-766,  if  it  be  an  equation  contain- 
ing linear  factors. 

767.  Divide  a  triangle  into  two  parts  having  equal 
perimeters  by  a  line  from  one  vertex. 

768.  Divide  a  parallelogram,  by  a  line  from  one  vertex, 
into  parts  whose  perimeters  shall  differ  by  a. 

769.  Take  equal  lengths  from  two  sides  of  a  triangle, 
such  that  the  sum  of  the  remainders  shall  equal  the  third 
side. 

770.  Draw  parallels  at  equal  distances,  respectively, 
from  the  sides  of  a  given  rectangle,  so  that  they  shall 
form  another  rectangle  of  given  perimeter. 

771.  Divide  one  side  of  a  triangle  into  two  parts  such 
that  their  difference  shall  be  equal  to  one  third  the  sum 
of  the  other  two  sides. 

772.  Find  a  point  E  in  the  same  line  with  the  four 
collinear  points  whose  order  is  A,  B,  C,  D,  such  that 
AE :  BE  =  BE  :  CE. 

773.  Produce  a  given  chord  of  a  circle,  so  that  the  tan- 
gent drawn  from  its  extremity  shall  have  a  given  length. 

774.  Construct  a  rectangle,  having  given  one  side  and 
the  sum  of  the  diagonal  and  the  other  side. 

775.  Divide  a  given  line  into  two  segments  such  that 
the  difference  of  their  squares  shall  equal  a2. 

776.  Divide  a  given  line  into  two  segments  such  that 
their  ratio  shall  be  equal  to  that  of  two  given  squares. 


112  MISCELLANEOUS   EXERCISES 

777.  In  the  triangle  ABC  draw  a  line  BE  parallel  to 
BC,  so  that  BE2  =  AC  -  CE. 

778.  In  the  triangle  ABC  draw  a  line  DE  parallel  to 
BC,  so  that  BE2  =  AB  •  CE. 

779.  In  the  triangle  ABC  draw  a  line  BE  parallel  to 
BC,  so  that  the  perimeter  of  the  triangle  ABE  shall  equal 
half  the  perimeter  of  the  trapezoid  BCEB. 

780.  Inscribe  in  a  given  triangle  a  rectangle  with  a 
given   perimeter. 

781.  Construct  a  right  triangle,  given  the  perimeter 
and  the  altitude  on  the  hypotenuse. 

782.  In  a  given  square  inscribe  five  equal  circles,  so 
that  the  middle  circle  shall  touch  the  other  four,  and  each 
of  these  four  shall  touch  two  adjacent  sides  of  the  square. 

783.  Given  a  rectangle  ;  construct  a  square  such  that 
the  perimeters  of  the  two  figures  shall  have  the  same  ratio 
as  their  areas. 

784.  In  a  triangle  ABC  draw  a  line  BE  between  AC 
and  BC,  so  that  the  triangle  CBE  shall  be  isosceles  and 
equal  in  area  to  half  the  triangle  ABC. 

785.  Within  a  given  circle  construct  five  equal  squares, 
so  that  each  of  the  four  outer  squares  shall  have  two  ver- 
tices in  the  circumference,  and  one  side  in  common  with 
the  fifth  square. 

786.  Construct  a  square  which  shall  be  to  a  given 
square  as  3  :  2. 

787.  Construct  two  lines,  given  their  ratio  and  their 
product. 

788.  Construct  two  lines,  given  their  ratio  and  the  dif- 
ference of  their  squares. 


MISCELLANEOUS   EXERCISES  113 

789.  Divide  a  given  trapezoid  into  two  equivalent  parts 
by  a  line  parallel  to  the  bases. 

790.  Divide  a  given  line  into  two  parts,  such  that  the 
smaller  shall  be  a  mean  proportional  between  the  greater 
and  the  difference  of  the  two. 

791.  Given  a  circle  and  a  chord ;  produce  the  chord  to 
a  point  At  such  that  the  tangent  from  A  shall  be  equal  to 
the  chord. 

792.  In  the  triangle  ABO  draw  DE  parallel  to  AB,  so 
that  AB.  AD  =  DE- CD. 

793.  Through  a  given  point  within  a  given  circle  draw 
a  chord  such  that  the  difference  of  the  two  segments  shall 
equal  a  given  line. 

794.  Divide  a  given  line  into  two  parts  so  that  the 
square  of  one  part  shall  equal  half  the  square  of  the  other. 

795.  Divide  a  given  line  into  two  parts  whose  product 
shall  equal  the  square  of  the  given  line. 

796.  Inscribe  in  a  given  equilateral  triangle  another 
equilateral  triangle  half  as  large. 

797.  Given  the  point  0  in  AB ;  find  a  point  D  in  AB 
between  B  and  <7,  such  that  AD2  =  BD  •  CD. 

798.  Construct  a  rectangle  equal  in  area  to  a  given 
rectangle,  having  a  perimeter  equal  to  the  perimeter  of 
another  given  rectangle. 

799.  Construct  a  right  triangle,  given  the  hypotenuse 
and  the  sum  of  the  legs. 

800.  Transform  a  given  square  into  an  isosceles  tri- 
angle in  which  the  sum  of  the  base  and  the  altitude  is 
given. 

conant's  ex.  geom.  —  8 


VII.     THEOREMS    AND    PROBLEMS    RELATING 
TO   THE   COMMON  GEOMETRICAL   SOLIDS 

801.  What  is  the  weight  of  a  ball  of  lead  whose  diam- 
eter is  1.68  m.,  the  specific  gravity  of  lead  being  11.36? 

802.  How  many  square  meters  of  surface  must  be 
cemented  in  constructing  a  cubical  reservoir  which  will 
hold  10,000  Kg.  of  water  ? 

803.  How  many  square  meters  of  surface  must  be 
cemented  in  constructing  a  reservoir  in  the  shape  of  a 
cylinder  15  m.  deep,  capable  of  holding  10,000,000  liters 
of  water? 

804.  A  cubical  vessel  requires  245  sq.  ft.  of  lead  for 
lining  the  bottom  and  sides ;  how  many  gallons  of  water 
will  it  hold? 

805.  What  is  the  volume  of  a  right  hexagonal  prism 
whose  height  is  8  ft.,  each  side  of  the  hexagon  being  6  ft.  ? 

806.  How  many  cubic  inches  of  mahogany  will  it  take 
to  veneer  the  top  of  a  circular  table  whose  diameter  is 
2  ft.  6  in.,  the  veneer  being  ^  of  an  inch  thick? 

807.  How  many  cubic  yards  of  stone  are  needed  to  build 
a  dam  1200  ft.  long,  40  ft.  high,  16  ft.  wide  at  the  bottom 
and  4  ft.  wide  at  the  top  ? 

808.  If  the  atmosphere  extends  to  a  height  of  45  miles 
above  the  surface  of  the  earth,  what  is  the  ratio  of  its 
volume  to  the  volume  of  the  earth,  assuming  the  latter  to 
be  a  sphere  whose  diameter  is  7912  miles  ? 

114 


COMMON    GEOMETRICAL   SOLIDS  115 

809.  How  much  will  it  cost  to  dig  a  well  6  ft.  in  diameter 
and  28  ft.  deep,  at  13.50  per  cubic  yard  of  earth  removed? 

810.  How  many  cubic  yards  of  earth  must  be  removed 
in  constructing  a  semicircular  tunnel  500  yd.  long,  the 
radius  being  8  ft.  ? 

811.  A  cylindrical  glass  jar  holds  1500  cu.  cm.  ;  find  its 
dimensions  if  its  depth  is  three  times  the  radius  of  its 
base. 

812.  An  iron  cylinder  6  in.  in  diameter  and  20  ft.  long 
is  reduced  in  diameter  in  a  lathe  half  an  inch;  what  is 
the  loss  in  weight,  the  specific  gravity  of  iron  being  7.2? 

813.  The  piston  of  a  pump  is  8  cm.  in  diameter,  and  the 
stroke  of  the  pump  is  50  cm. ;  how  many  liters  of  water 
are  pumped  out  by  500  strokes? 

814.  How  many  feet  of  wire  -fa  of  an  inch  in  diameter 
can  be  drawn  out  of  2  cu.  in.  of  brass? 

815.  Given  the  circumference  c  of  the  base  of  a  right 
cylinder,  and  the  total  surface  T;  find  the  volume. 

816.  Given  the  volume  V  of  a  right  cylinder,  and  the 
altitude  h ;  find  the  total  surface. 

817.  A  cone  whose  altitude  is  8  ft.  has  a  base  contain- 
ing 64  sq.  ft.  ;  at  what  distance  from  the  vertex  must  a 
plane  parallel  to  the  base  be  passed  to  contain  45  sq.  ft.  ? 

818.  A  pyramid  6  ft.  high  has  a  square  base  2.5  ft.  on  a 
side ;  find  the  area  of  a  section  made  by  a  plane  parallel 
to  the  base  and  2  ft.  from  the  vertex. 

819.  The  base  of  a  regular  pyramid  is  a  hexagon  which 
measures  2  ft.  on  a  side ;  find  the  altitude  of  the  pyramid 
if  the  lateral  area  is  six  times  the  area  of  the  base. 


116  COMMON   GEOMETRICAL    SOLIDS 

820.  Find  the  volume  of  a  regular  triangular  pyramid 
whose  altitude  is  18  in.  and  whose  base  edges  are  each 
5  in. 

821.  Find  the  total  surface  of  a  regular  triangular 
pyramid  if  each  side  of  its  base  is  4  ft.,  and  its  slant  height 
is  8  ft. 

822.  Find  the  altitude  of  a  triangular  pyramid  if  its  vol- 
ume is  26  cu.  ft.,  and  the  sides  of  its  base  are  3,  4,  and  5  ft. 

823.  A  regular  pyramid  with  a  square  base  contains 
122  cu.  yd.,  and  its  altitude  is  41  ft. ;  what  is  the  area  of 
its  base  ? 

824.  The  total  surface  of  a  regular  quadrangular  pyra- 
mid is  jT,  and  its  eight  edges  are  equal,  each  to  each ; 
find  the  length  of  an  edge. 

825.  The  radius  of  the  base  of  a  cone  of  revolution  is  30 
in.,  and  its  altitude  is  6  ft. ;  how  far  from  its  base  must  a 
parallel  plane  be  passed  to  cut  a  section  of  the  cone  whose 
radius  shall  be  20  in.  ? 

826.  The  slant  height  of  a  cone  is  3  in. ;  how  must  the 
slant  height  be  divided  in  order  that  the  lateral  surface 
may  be  divided  into  two  equal  parts?  into  three  equal 
parts? 

827.  The  volume  of  a  cone  is  V;  what  does  the  volume 
become  if  the  altitude  is  doubled?  if  the  radius  of  the 
base  is  doubled?  if  both  are  doubled?  if  both  are  trebled? 

828.  A  conical  mound  of  earth  measures  148  yd.  around 
the  base,  and  its  slope  measures  52  yd. ;  how  many  cubic 
yards  are  there  in  the  mound  ? 

829.  The  weight  of  a  cone  of  revolution  of  silver  is  5 
Kg.,  and  its  altitude  is  twice  the  diameter  of  its  base ;  find 


COMMON   GEOMETRICAL   SOLIDS  117 

the  dimensions  of  the  cone,  the  specific  gravity  of  silver 
being  10.47. 

830.  Find  the  volume  of  a  cone  of  revolution  whose  slant 
height  is  equal  to  the  diameter  of  its  base,  the  total  surface 
being  T. 

831.  Find  the  lateral  surface  of  the  frustum  of  a  right 
cone  whose  altitude  is  2  m.,  and  the  radii  of  whose  bases 
are  6  m.  and  3  m.  respectively. 

832.  The  altitude  of  a  cone  of  revolution  is  8  in.  and  the 
radius  of  its  base  is  2  in.  ;  find  the  lateral  surface  of  the 
frustum  made  by  a  plane  parallel  to  the  base  3  in.  from  the 
vertex. 

833.  A  bucket  is  14  in.  in  diameter  at  the  top  and  10  in. 
in  diameter  at  the  bottom,  and  its  depth  is  12  in. ;  how 
many  gallons  will  it  hold? 

834.  The  bases  of  a  frustum  of  a  regular  pyramid  are 
hexagons  whose  sides  are  1  ft.  and  2  ft.  respectively,  and 
its  volume  is  14  cu.  ft. ;  find  its  altitude. 

835.  The  radii  of  the  bases  of  a  right  frustum  are  3  m. 
and  5  m.  respectively,  and  its  volume  is  30  cu.  m. ;  find  the 
volume  of  the  cone  from  which  the  frustum  was  cut. 

836.  The  frustum  of  a  cone  of  revolution  is  12  ft.  high 
and  its  volume  is  228  cu.  ft.  ;  find  the  radii  of  the  bases 
if  their  sum  is  6  ft. 

837.  A  square  tower  is  built  14  m.  high,  each  side  of  the 
base  being  15  m.,  and  each  side  of  the  top  being  11  m. 
Within  the  tower  is  a  circular  shaft  2  m.  in  diameter,  for 
a  spiral  staircase.  How  many  cubic  meters  of  material 
were  used  in  building  the  tower  ? 

838.  What  is  the  locus  of  tangents  drawn  from  a  point 
to  a  sphere  ? 


118  COMMON    GEOMETRICAL   SOLIDS 

839.  What  is  the  locus  of  all  points  3  in.  from  the  sur- 
face of  a  sphere  whose  radius  is  5  in.  ?  6  in.  from  the 
surface  ?  5  in.  from  the  surface  ? 

840.  The  distances  from  the  poles  of  a  small  circle  of  a 
sphere  to  the  circumference  of  the  circle  are  4  m.  and 
7  m.  respectively  ;  find  the  area  of  the  circle. 

841.  Find  the  surface  of  a  lune  if  its  angle  is  31°  30', 
and  the  radius  of  the  sphere  is  2  ft. 

842.  The  angles  of  a  spherical  triangle  are  38°  41', 
84°  50',  115°  52',  and  the  radius  of  the  sphere  is  8  in. ; 
find  the  area  of  the  triangle. 

843.  The  sides  of  a  spherical  triangle  are  52°,  83°,  116°, 
and  the  radius  of  the  sphere  is  7  in.  ;  find  the  area  of  its 
polar  triangle. 

844.  The  diameter  of  a  sphere  is  16  in. ;  find  the  area 
of  a  zone  whose  altitude  is  3  in. 

845.  In  a  sphere  of  radius  r,  find  the  altitude  of  a  zone 
whose  area  is  equal  to  that  of  a  great  circle.  If  the  area 
is  doubled,  is  the  altitude  doubled  ? 

846.  The  radius  of  a  sphere  is  6  in.  The  sphere  is  cut 
by  two  parallel  planes  respectively  2  ft.  and  4  ft.  from 
the  center.  Find  the  area  of  the  zone  thus  formed.  (Two 
solutions.) 

847.  Find  the  area  of  a  zone  on  a  sphere  of  radius  r 
illuminated  by  a  lamp  placed  at  a  distance  a  from  the 
sphere. 

848.  How  far  from  the  surface  of  a  sphere  must  the  eye 
be  placed  to  see  one  sixth  of  the  surface  ? 

849.  Find  the  volume  of  a  sphere  if  the  section  made 
by  a  plane  3  cm.  from  the  center  contains  12  sq.  cm. 


COMMON   GEOMETRICAL   SOLIDS  119 

850.  How  many  cubic  inches  of  iron  are  there  in  a  shell 
|  in.  thick,  if  the  inside  diameter  of  the  shell  is  7|  in.  ? 

851.  If  an  iron  ball  4  in.  in  diameter  weighs  9  lb., 
what  is  the  weight  of  a  hollow  iron  shell  J  in.  thick,  its 
outside  diameter  being  12  in.  ? 

852.  Find  the  volume  of  a  triangular  spherical  pyramid 
if  the  angles  at  its  base  are  each  110°,  and  the  radius  of 
the  sphere  is  8  in. 

853.  The  radius  of  the  base  of  a  segment  of  a  sphere  is 
14  in.,  and  its  height  is  8  in.  ;  find  its  volume. 

854.  A  sphere  of  radius  8  ft.  is  cut  by  parallel  planes 
on  the  same  side  of  the  center,  distant  from  the  center 
3  ft.  and  5  ft.  respectively ;  find  the  volume  comprised 
within  the  planes. 

855.  A  spherical  segment  is  half  as  large  as  the  spheri- 
cal sector  to  which  it  belongs ;  if  the  radius  of  the  sphere 
is  4  in.,  find  the  altitude  of  the  segment. 

856.  Dry  sand  is  poured  over  a  sphere  of  2  ft.  radius 
until  it  forms  a  conical  pile  whose  height  is  8  ft.,  and  the 
circumference  of  whose  base  is  50  ft. ;  how  many  cubic 
feet  of  sand  are  used  ? 

857.  Find  the  dimensions  of  a  right  circular  cone  1  in. 
high  that  can  be  made  from  a  cubic  inch  of  material. 

858.  A  regular  hexagonal  pyramid  is  inscribed  in  a  cone 
of  revolution ;  the  slant  height  of  the  pyramid  is  12  ft. 
and  its  lateral  edges  are  each  13  ft.  ;  how  many  cubic  feet 
of  the  cone  are  not  comprised  in  the  pyramid  ? 

859.  Find  the  area  of  a  zone  generated  by  an  arc  of  20° 
revolving  about  a  diameter  passing  through  one  of  its 
extremities,  the  radius  being  a. 


120  COMMON   GEOMETRICAL   SOLIDS 

860.  A  sphere  of  radius  4  in.  is  cut  by  two  parallel 
planes  equally  distant  from  the  center,  so  that  the  area  of 
the  zone  between  them  is  equal  to  the  sum  of  the  areas  of 
the  sections  made  by  the  planes  ;  find  the  distance  of  either 
plane  from  the  center. 

861.  Prove  that  a  line  and  a  plane  which  are  perpen- 
dicular to  the  same  straight  line  are  parallel. 

862.  If  two  planes  are  parallel,  a  line  parallel  to  one  of 
them  through  a  point  in  the  other  lies  in  the  other. 

863.  The  alternate  interior  dihedral  angles  formed  by 
the  intersection  of  two  parallel  planes  with  a  third  plane 
are  equal. 

864.  If  a  plane  be  drawn  through  a  diagonal  of  a  paral- 
lelogram, the  perpendiculars  to  it  from  the  extremities  of 
the  other  diagonal  are  equal. 

865.  If  each  of  two  intersecting  planes  be  cut  by  two 
parallel  planes  not  parallel  to  the  intersection  of  the  first 
two  planes,  the  intersections  of  these  planes  with  the  paral- 
lel planes  are  the  edges  of  equal  dihedral  angles. 

866.  The  three  planes  bisecting  the  dihedrals  of  a  tri- 
hedral meet  in  a  common  straight  line. 

867.  If  from  any  point  in  either  face  of  a  dihedral  a  line 
be  drawn  perpendicular  to  the  edge  of  the  dihedral,  and 
from  the  same  point  a  perpendicular  be  dropped  upon  the 
other  face  of  the  dihedral,  the  plane  of  these  two  lines 
is  perpendicular  to  the  plane  containing  the  point  from 
which  they  are  drawn. 

868.  The  volume  of  a  regular  pyramid  is  equal  to  its 
lateral  area  multiplied  by  one  third  the  distance  from  the 
center  of  its  base  to  any  lateral  face. 


COMMON   GEOMETRICAL   SOLIDS  121 

869.  Lines  joining  the  middle  points  of  two  pairs  of 
opposite  edges  of  a  tetrahedron  inclose  a  parallelogram. 

870.  A  plane  passed  through  the  center  of  a  parallelo- 
piped  divides  the  parallelopiped  into  two  solids  which  are 
equal  in  volume.  Are  the  solids  equal  in  all  respects  ? 
Are  they  symmetrical  ? 

871.  How  many  cubic  feet  of  earth  and  rock  are  taken  out 
by  a  boring  3  in.  in  diameter  sunk  2400  ft.  into  the  ground? 

872.  Two  tetrahedrons  are  equal  in  all  respects  if  three 
faces  of  the  one  are  equal  respectively  to  three  faces  of  the 
other,  and  are  similarly  placed. 

873.  The  volume  of  a  triangular  prism  is  equal  to  a 
lateral  face  multiplied  by  one  half  the  distance  from  any 
point  in  the  opposite  lateral  edge  to  that  face. 

874.  The  volume  of  a  truncated  right  parallelopiped  is 
equal  to  the  area  of  its  lower  base  multiplied  by  the  per- 
pendicular drawn  to  the  lower  base  from  the  center  of  the 
upper  base. 

875.  The  perpendicular  drawn  to  the  lower  base  of  a 
truncated  right  triangular  prism  from  the  intersection  of 
the  medians  of  the  upper  base  is  equal  to  one  third  the 
sum  of  the  lateral  edges. 

876.  The  three  planes  passing  through  the  lateral  edges 
of  a  triangular  pyramid,  bisecting  the  sides  of  the  base, 
intersect  in  one  straight  line. 

877.  Any  section  of  a  tetrahedron  parallel  to  two  oppo- 
site edges  of  the  tetrahedron  is  a  parallelogram. 

878.  The  sum  of  the  squares  of  the  four  diagonals  of  a 
parallelopiped  is  equal  to  the  sum  of  the  squares  of  the 
twelve  edges. 


122  COMMON   GEOMETRICAL   SOLIDS 

879.  A  section  of  a  tetrahedron  ABCD  containing  CD 
and  perpendicular  to  AB  intersects  the  faces  ABC  and 
ABD  in  CE  and  ED  respectively.  If  the  bisector  of  the 
angle  CED  meets  CD  in  F,  prove  that  CF :  DF  =  area 
ABC:  area  ABD. 

880.  The  radius  of  the  base  of  a  right  circular  cone  of 
copper  is  4  in.  and  its  altitude  is  10  in.;  a  core  1  in.  in 
diameter  is  bored  out,  whose  axis  coincides  with  the  axis 
of  the  cone ;  what  is  the  weight  of  the  copper  removed, 
its  specific  gravity  being  8.75  ? 

881.  If  the  four  diagonals  of  a  quadrangular  prism  are 
concurrent,  the  prism  is  a  parallelopiped. 

882.  ABC  and  A'B'C  are  two  polar  triangles  on  a 
sphere  whose  center  is  0  ;  prove  that  0A!  is  perpen- 
dicular to  the  plane  of  OBC. 

883.  Any  angle  of  a  spherical  triangle  is  greater  than 
the  difference  between  180°  and  the  sum  of  the  other  two 
angles. 

884.  The  distance  between  the  centers  of  two  spheres 
whose  radii  are  25  in.  and  17  in.  respectively,  is  28  in.  ; 
what  is  the  diameter  of  the  circle  of  intersection  of  the 
two  spheres? 

885.  The  volume  of  a  quadrangular  spherical  pyramid 
whose  base  angles  are  110°,  122°,  135°,  and  146°  respec- 
tively, is  15  cu.  ft.;  what  is  the  volume  of  the  entire 
sphere  ? 

886.  A  spherical  cannon  ball  8  in.  in  diameter  is 
dropped  into  a  cubical  tank  full  of  water.  If  the  inside 
edges  of  the  tank  are  each  8  in.,  how  many  cubic  inches  of 
water  will  remain  in  it  after  the  cannon  ball  lias  been 
dropped  in? 


COMMON   GEOMETRICAL   SOLIDS  123 

887.  How  many  spherical  bullets,  each  j  in.  in  diameter, 
can  be  run  from  a  cylindrical  block  of  lead  7  in.  in  diam- 
eter and  18  in.  long? 

888.  The  volume  of  a  cylinder  of  revolution  is  equal  to 
the  area  of  its  generating  rectangle  multiplied  by  the  cir- 
cumference of  a  circle  whose  radius  is  the  distance  from 
the  center  of  the  rectangle  to  the  axis  of  the  cylinder. 

889.  The  volume  of  a  cone  of  revolution  is  equal  to  its 
lateral  area  multiplied  by  one  third  the  distance  of  any 
element  of  its  lateral  surface  from  the  center  of  the  base. 

890.  A  cone  of  revolution  is  circumscribed  about  a 
sphere  whose  diameter  is  two  thirds  the  altitude  of  the 
cone ;  prove  that  the  lateral  surface  and  the  volume  of 
the  cone  are  respectively  three  halves  and  nine  fourths 
the  surface  and  the  volume  of  the  sphere. 

891.  A  square  whose  area  is  m  revolves  about  one  of  its 
diagonals  as  an  axis;  find  the  volume  and  the  convex 
surface  of  the  solid  thus  generated. 

892.  A  right  triangle  whose  legs  are  6  in.  and  8  in.  respec- 
tively, revolves  about  its  hypotenuse  as  an  axis ;  find  the 
volume  and  the  convex  surface  of  the  solid  thus  generated. 

893.  An  equilateral  triangle  whose  side  is  a  revolves 
about  one  of  its  sides  as  an  axis  ;  find  the  volume  and 
area  of  the  surface  of  the  solid  generated. 

894.  Find  the  lateral  area  and  the  volume  of  a  cylinder 
of  revolution  whose  altitude  is  equal  to  the  diameter  of 
its  base,  inscribed  in  a  sphere  whose  radius  is  r. 

895.  An  equilateral  triangle  whose  side  is  a  revolves 
about  a  line  through  one  of  its  vertices  and  parallel  to  the 
opposite  side;  find  the  lateral  area  and  the  volume  of  the 
solid  thus  generated. 


124  COMMON   GEOMETRICAL   SOLIDS 

896.  The  cross  section  of  a  tunnel  2|  mi.  long  is  a 
rectangle  6  yd.  long  and  2  yd.  in  height,  surmounted  by 
a  semicircle  whose  diameter  equals  the  length  of  the  rec- 
tangle ;  how  many  cubic  yd.  of  material  were  removed 
in  constructing  it? 

897.  The  volume  of  a  cone  of  revolution  equals  the 
area  of  its  generating  triangle  multiplied  by  the  circum- 
ference of  a  circle  whose  radius  is  the  distance  from  the 
orthocenter  of  the  triangle  to  the  axis  of  the  cone. 

898.  If  the  earth  be  a  sphere  of  radius  r,  what  is  the 
area  of  a  zone  visible  from  a  point  whose  height  above 
the  surface  of  the  earth  is  A? 

899.  A  projectile  consists  of  a  cylinder  of  revolution 
18  in.  long  and  a  cone  of  revolution  12  in.  long  ;  if  the 
diameter  of  the  projectile  be  13  in.,  what  is  its  volume? 

900.  Find  the  surface  of  a  sphere  circumscribing  a 
regular  tetrahedron,  one  of  whose  edges  is  8  in. 


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Shakespeare's  Merchant  of  Venice.  Professor  Felix  E.  Schelling, 
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Shakespeare's  Julius  Caesar.  Dr.  Hamilton  W.  Mabie,  "The 
Outlook."     35  cents. 

Shakespeare's  Macbeth.  Professor  T.  M.  Parrot,  Princeton 
University.     40  cents. 

Milton's  Minor  Poems.  Professor  Mary  A.  Jordan,  Smith 
College.     35   C3nts. 

Addison's  Sir  Roger  de  Coverley  Papers.  Professor  C.  T.  Win- 
chester, Wesleyan  University.     40  cents. 

Goldsmith's  Vicar  of  Wakefield.  Professor  James  A.  Tufts, 
Phillips  Exeter  Academy.     45  cents. 

Burke's  Speech  on  Conciliation.  Professor  William  MacDonald, 
Brown  University.     35  cents. 

Coleridge's  The  Ancient  Mariner.  Professor  Geo.  E.  Wood- 
berry,  Columbia  University.     30  cents. 

Scott's  Ivanhoe.  Professor  Francis  H.  Stoddard,  New  York 
University.     50  cents. 

Scott's  Lady  of  the  Lake.  Professor  R.  M.  Alden,  Leland  Stan- 
ford Jr.  University.     40  cents. 

Macaulay's  Milton.  Rev.  E.  L.  Gulick,  Lawrence ville  School. 
35  cents. 

Macaulay's  Addison.  Professor  Charles  F.  McClumpha,  Uni- 
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Carlyle's  Essay  on  Burns.  Professor  Edwin  Mims,  Trinity 
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George  Eliot's  Silas  Marner.  Professor  W.  L.  Cross,  Yale 
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Tennyson's  Princess.  Professor  Katharine  Lee  Bates,  Wellesley 
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Macaulay's  Life  of  Johnson.  Professor  J.  Scott  Clark,  North- 
western University.     35  cents. 


AMERICAN    BOOK    COMPANY 

[99] 


WALKER'S    ESSENTIALS    IN 
ENGLISH   HISTORY 

FROM  THE   EARLIEST 

RECORDS 
TO  THE  PRESENT  DAY 

By  Albert  Perry  Walker,  A.M.,  Master  in  History 

English  High  School,  Boston 

In  consultation  with  Albert  Bushnell  Hart,  LL.D. 

Professor  of  History,  Harvard  University 

Half-leather,  8vo.,  592  pages,  with  maps  and  illustrations 

Price,  $1.50 


THIS  volume  was  prepared  for  a  year's  work  in  the  high 
school.  The  book  is  admirably  adapted  for  this  use,  for  it 
is  a  model  of  good  historical  exposition,  unusually  clear  in 
expression,  logical  and  coherent  in  arrangement,  and  accurate  in 
statement.  The  essential  facts  in  the  development  of  the  British 
Empire  are  vividly  described,  and  the  relations  of  cause  and  effect 
are  clearly  brought  out.  The  text  meets  thoroughly  the  most  ex- 
acting college  entrance  requirements.  The  narrative  follows  the 
chronological  order,  and  is  full  of  matter  which  is  as  interesting  as 
it  is  significant,  ending  with  a  masterly  summary  of  England's  con- 
tribution to  civilization. 

The  numerous  maps  give  to  every  event  its  local  habitation, 
and  the  pictures  are  reproductions  of  real  objects,  and  serve  not 
only  to  vivify  the  text  but  also  to  stimulate  the  historical  imagination. 
At  the  close  of  each  chapter  are  inserted  lists  of  carefully  prepared 
topics  for  intensive  study,  and  of  brief  selections  for  collateral  or 
supplementary  reading.  In  the  appendix  is  given  a  brief  bibli- 
ography for  a  working  school  library,  as  well  as  a  more  compre- 
hensive bibliography  of  English  history.  The  volume  closes  with 
an  excellent  index. 


American   Book  Company 

NEW  YORK  CINCINNATI  CHICAGO 

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